Is quantizing acceleration equivalent to quantizing gravity? It's been said that gravity is locally equivalent to acceleration.  It is also been said, that we need to quantize gravity. But doesn't that imply that this would have to be equivalent to quantizing acceleration? 
 A: No, that betrays a misunderstanding of what "quantizing" means. 
The usual definition of quantization is the conversion of continuous values to discrete values, but that's not what it means in physics. In physics, quantization is the process of turning a classical theory into a quantum one, which is a far more subtle process. And once you do have a quantum theory, not everything in that theory needs to be discrete. For example, electric charge is discrete, but mass isn't; the energy of an electron bound to a proton is discrete, but the energy of a free electron isn't. Saying that quantum mechanics is just classical mechanics with discrete values is like saying coding is just writing in English with a monospaced font.
The problem with quantizing gravity is more specific. It has nothing to do with quantizing particles in a classical gravitational field, which might be what you mean by quantizing acceleration; we already know how to do that perfectly well. (It also does not lead to discrete values of acceleration.) The problem is giving the gravitational field itself quantum dynamics. 
You might think this is unnecessary because gravitational fields are "fictitious", as they are "actually" just accelerations by the equivalence principle. But that's getting the equivalence principle backwards. The point is that a gravitational field is only locally equivalent to an acceleration, but a position-dependent gravitational field isn't. These tidal effects are described in terms of the curvature of spacetime in general relativity, which is physical and cannot be regarded as fictitious.
One might also think that we could treat spacetime classically while quantizing everything else. But Einstein's equations relate the stress-energy tensor (of quantum matter) to the curvature. If the stress-energy tensor can be in superposition, and the curvature can't, how can this make sense?
Once we do give the gravitational field quantum dynamics, the problem is that gravitational interactions grow stronger at high energies. In order to correctly describe the behavior of gravitons, we have to allow additional terms in Einstein's equations as the energy goes up. When the energy reaches the Planck energy, infinitely many terms become important simultaneously. Since we don't know how big these terms are, we can't make any predictions at this scale. We need an ultraviolet completion, a larger theory that determines all of these terms at once. That's the kind of thing a quantum gravity theorist is looking for.
