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While it's common to describe gravity as a fictitious force we should be cautious about the use of the adjective fictitious as this is a technical term meaning the gravitational force is not fundamental but is the result of an underlying property. The force itself most certainly exists as anyone who has been sat on by an elephant can attest.

There is a sense in which all forces are fictitious since they are all the emergent long range behaviour of quantum fields, so gravity is not unique in this respect. For more on this see Can all fundamental forces be fictitious forces?

Anyhow, the object responsible for the gravitational force is a tensor field called the metric, and when we quantise gravity we are quantising the metric not the force. The graviton then emerges as the excitation of the quantum field that describes the metric. As with other quantum fields we can have real gravitons that are the building blocks of gravitational waves and virtual gravitons used in scattering calculations.

Gravity is not equivalent to an accelerated frame. It's locally equivalent to an accelerated frame. That means that a point-like observed will never be able to tell whether he/she is in a gravitational field or in an accelerating spaceship. But an observer that has some characteristic size will experience tidal forces. Tidal forces are a result of a non zero curvature of space. On the other hand going to an accelerated frame does not curve spacetime because it's just a change of variables.

The mathematical statement is that you can always find a change of coordinates which, at a given point, sets to zero the first derivatives of the metric. Namely you can make the Christoffel symbols $Gamma^mu_{nurho}$ vanish at a given point. On the other hand the second derivatives of the metric, which encode the curvature, cannot be set to zero.

There is a nice exercise that demonstrates the difference between accelerating frames and curved spacetime. Suppose you have two particles following two parallel geodesics. In curved spacetime the geodesics will not stay parallel. This is discussed in the book Schutz - A first course in general relativity at the end of Section 6.5. I'll summarize the punch line but I encourage you to check the derivation there.

Call $vec{xi}$ the vector that connects two geodesics $vec{V}$ and $vec{V}'$ initially parallel. In the book the following equation is proven
$$
nabla_Vnabla_V xi^alpha = R^alpha_{phantom{a}munubeta}V^mu V^nu xi^beta,.
$$
where $nabla_V = V^mu nabla_mu$, $nabla$ being the covariant derivative and $R^alpha_{phantom{a}munubeta}$ is the Riemann tensor, which measures the curvature. The equation above states that the variation of $vec{xi}$ along the geodesic is not zero but proportional to the curvature, thus showing that geodesics deviate from being parallel in curved space. This is a frame-independent effect and cannot be realized by going to an accelerated frame.

An observer with a characteristic size of the order of $1/sqrt{R^mu_{phantom{a}nurhobeta}}$ would be able to notice this effect.

None of the answers current explain an aspect of the question that I am interested in: what goes wrong if you try to construct a theory of physics where gravity is not quantized?

There are various arguments that strongly suggest that everything coupled to a quantum system should, fundamentally, also be quantum.

We know that the stress-energy tensor sources curvature for the gravitational field,
$$G_{munu} sim T_{munu}$$
but in a quantum theory the stress-energy tensor does not have a definite value, but rather may be in superposition. So then how do we describe the curvature? If you say the curvature may be in superposition too, so that $G_{munu} = T_{munu}$ holds for each branch of the superposition, then you've just quantized gravity -- quantization is exactly the process where we treat the set of classical physical states of a system as separate quantum states which may be superposed.

The only other option which reduces to the classical result when the matter is nearly classical is
$$G_{munu} sim langle T_{munu} rangle.$$
However, this is extremely strange for many reasons. For example, consider a particle of mass $m$ which is in an equal superposition of being here or in Andromeda. Then the classical gravitational field would be that of two masses $m/2$, each in one galaxy. If the particle is measured, the wavefunction collapses, and the gravitational field instantaneously changes, so the observed mass in Andromeda becomes either $m$ or zero. This nonlocal change in the field allows superluminal signalling by somebody in the Milky Way. (There's nothing special about gravity here; it would also hold if we insisted on a classical electromagnetic field. In either case, when the field is quantized, this problem is avoided by the usual way in quantum field theory.)

In addition, energy conservation may be violated. This is easier to see with the electromagnetic field. If one starts with an excited atom in an empty cavity, in state $|e rangle$, after some time it will be in the superposition $(|e rangle + |g rangle) / sqrt{2}$. If you insist the electromagnetic field have a definite classical configuration, then the branches of the wavefunction do not have equal energy. When you measure the energy, you'll generally find a different result than the initial energy; it can only match on average.

This is essentially the erroneous BKS theory which was rendered obsolete with the quantization of the electromagnetic field. In this case the wavefunction is $(|erangle otimes |0 rangle + |g rangle otimes |1 rangle) / sqrt{2}$ where the second factor indicates the number of photons, and the two branches of the wavefunction have exactly the same energy as they must. Similarly, if one couples to classical gravity, one must allow violations of energy conservation that only cancel out on the average, but there's no problem for quantized gravity.

I'm sure the mathematicians can come up with more sophisticated, complicated reasons that classical and quantum theories don't mesh, but these immediate issues are already bad enough.

You could also ask why the photon is necessary, if electromagnetism is a classical force based on Yang-Mills fields with gauge group U(1). Or also, why the gluons, the W, Z AND the Higgs boson are necessary, since non-abelian Yang-Mills fields are also meaningful as classical fields. In my opinion, the answer to this question, and why fields are to be quantized, must include two subtle issues:

1. Quanta are not fundamental, but, as previous questions remark, are excitations from vacuum of certain FIELDS on space-time. What is relevant is the quantization of the action, that generally implies the quantization of energy and other magnitudes like angular momentum.


2. Gravity has a different status with respect to other forces due to its universality, not due to it being a "pseudo-force". Gravity couples to everything, while other fields couple to certain properties of space-time like electric (magnetic) charge, flavor or color.

Moreover, the question of the need of the quantization of the gravitational field is evident when seeing the Einstein field equations for gravity: one side is the matter-energy having mass, energy, and quantum numbers, the other side is the geometry or metric of space-time. If identical, well, one should wonder if the metric itself has these features. String theory or loop quantum gravity show differently how the space-time itself could handle with quantum numbers. The problem with quantum gravity is not that we don't need gravitons. Indeed, Newton's gravity itself imply certain field theory in the form of Poisson equation that Einstein himself used as model to reproduce an analogy for building up his equations for gravity. The problem with quantum gravity and gravitons is in the heart of your question: if we model space-time like a metric and geometry, why do we need gravitons? We need gravitons because they must be there. Quantum theory is correct, even if some day is proved to be uncomplete or it must be modified to include gravity. Maxwell's equations are superseded by QED and the electroweakt theory at high energies, there new particles appear: the W, Z bosons and the Higgs (for consistency). Conceptually, maybe, the issue is understando how a set of gravitons could determine the geometry of the metric? No, the issue with gravitons is that General Relativity in a canonical quantum theory behaves badly. Calculations diverge. By the other hand, the space-time metric, the one in General Relativity, can not be the whole story...Just we know the Standard Model is not the whole story...The spacetime metrics in some concrete circumstances also diverge CLASSICALLY! Every theoretical physicist know that space-time singularities are a problem in most of the classical theories of gravity. You get singularities in black holes (hidden under event horizon, due to the cosmic censorship hypothesis), and you get singularities at the beginning of the time...In both cases, you have a very dense object in a very tiny space. Such extreme density conditions make us think that General Relativity and the description of space-time with a metric is only an approximation or a very good model excepting extreme cases (black holes, the Big Bang,...or similar). There, enter quantum gravity and gravitons. Graviton scattering must domine in such regime or produce some kind of extreme "matter"/object whose description with a metric is bad. Of course, some people work on the idea that black holes and space-time is some kind of "condensate" of gravitons or superfluid made of some preonic substance yet to discover (the nature of the microstates of black holes is only approached in some extreme cases with superstring theory).
In summary:

1) A graviton is necessary due to universality description of all the forces as interchanging force carriers.

2) A graviton is necessary since we believe graviton excitations, maybe Wheeler's space-time foam in some form or alike, must dominate the description of very dense objects (microscopic black holes, the beginning of the time, and other similar examples as space-time singularities).

However, graviton scattering behaves badly in general relativity. Taking a conservative canonical quantum gravity approach provides divergent results. Only string theory and loop quantum gravity, and some minor third ways to quantum gravity, shed light on how to calculate these divergences.
String theory provides a unifying framework to handle with all the "fundamental forces" and matter field. However, after two revolutions, and no hints of extra dimensions in experiments and detectors (and a critical 4D value from gravitational wave observations to date), we have no evidence from strings or p-branes yet. Loop quantum gravity (a modification of the canonical quantum gravity approach) provides an example of the quantization of geometry using a different technique than that in string theory. Area and volume are quantized in LQG. What are gravitons then? Gravitons in string theory are certain kind of excitations of the fundamental string (or brane). This fact is also remarked in the emergence of a symmetrical tensor when calculating the excitations of the string from the "vacuum". Gravitons in LQG are more subtle, I imagine them like polymer-like excitations from the area and volume operators, derived from spin networks and other discrete structures of the theory (I am not expert on that field, so I am being imprecise quite likely...).

3) Gravitons, photons, Higgs bosons, gluons, are likely not fundamental...Why do we need them? Because quantum fields can be represented as entities whose excitations produce particles. It happens with fermions as well. There is only a single electron field in all the Universe. However, the excitations in that field are the electrons we observe, reverberation of the beginning of the time...Just like gold atoms are produced in supernovae, electrons (or quarks) in the Universe were produced in the farthest past, and what remains is a rest from the annihilation with vacuum billions of years ago.

Gravitons, like photons and other particles, were produced in the beginning of the time. We don't understand what happened there, when GRAVITON scattering was dominant since the temperature was so hot, and the density so high, that we can not neglect gravitational interactions, usually weak when present electromagnetic or nuclear forces, or negligible only when you are not in a place where you have dense matter in a tiny volume (microscopic AND heavy black holes). That is why we need to understand better gravitons. Before the discovery of gravitational waves, that by duality imply the existence of the gravitons, some people wondered if gravity should be quantized. I think that question is not (if ever it was) relevant now. Gravitational waves do exist and then, gravitons (in some form) may exist. But, this have nothing to do with the classical existence of gravity. Before the Quantum Mechanics, physicists discussed if light was a wave or a particle. Well, light is both! Why do we need PHOTONS? We need photons since without photons (quanta of light) we could not explain wavy the photoelectric effect or the black-body radiation. Indeed, you are all embedded in a cosmic microwave background of photons emitted by the Big Bang, with temperature about 2.73 K. We believe there are also a neutrino and a graviton background as well. So, we need gravitons as well to understand the Universe! We can not understand the beginning of the Universe without understanding gravitons and the quantum nature of gravity.

By the Heisenberg uncertainty principle, things smaller * higher energy than hbar*c must be described as waves, and this is obviously true for point singularities. Thus black holes fall not only in the domain of general relativity, but also quantum mechanics, which motivates the quest to quantize gravity. Quantum effects associated with gravity have been used for years, ie. Hawking radiation, but not in the full theoretic framework. People have tried to make gauge theories compatible with curved spacetime, for instance using the covariant derivative notation for minimal coupling similar to its use for curved space, but AFAIK, the problem with gravitons is that they are non-renormalizable. See https://arxiv.org/abs/gr-qc/0405033 for a nontraditional gauge theory based on the geometry of space.