Now that we have determined that classical gravity with quantized matter is inconsistent, we are led to quantum gravity, including the quantization of the metric. I'll leave the lengthy details for a text like Birrel and Davies or Wald, but suffice it to say you can kind of do this. The problem is that it is a technically challenging and mathematically difficult subject. Again, I highly recommend any and all to read the first 10 pages of B&D if you are serious about quantum gravity. It is subtle but necessary information to even begin to tackle this subject seriously.
In fact, it should IMO be *REQUIRED* reading to even be allowed to post on this board as anyone posting about strings, SUGRA or loops or any quantum gravity proposal who hasn't mastered this material, is in quite literally over there heads. Seriously, nothing makes sense if you haven't mastered the basics first, you simply cannot avoid it.
Anyway its important to note the following about these 'semiclassical' solutions:
1) The equivalence principle states that all forms of matter and energy must couple equally strongly to gravity, including gravitational self energy. Said another way, a graviton will feel gravity just as strongly as say a photon would. Likewise, the quantum vacuum can just as easily spit out a graviton as it can a photon. In other words, you have to consider everything at once and you can never find a limit where various quantities decouple smoothly. Said yet another way, the nonlinearity of gravity frustrates all attempts to ignore quantum gravity. This is of course nasty, b/c in principle we don't know what the lagrangian for all matter is, as well as being forced to consider the full backreaction of all metric variations onto itself.
Still, we can perform a trick. The trick is to take the small linearized graviton contribution and pull it into the matter source term. This is similar in spirit to looking at photon emmision from a background object (like an atom) immersed in a rapidly changing relativistic electric or magnetic field. This kind of works, so long as one keeps the backreaction small and controlled.
2) The effective field theory of quantum gravity requires an expansion around a small parameter in order to have a valid perturbative setup, as well as to keep the aforementioned backreaction small and under control. The only such number available is given by dimensional analysis and is epsilon = E^2/Mpl^2. Expansion around this quantity is well defined, so long as an energy or length scale is chosen that forces it to be smaller than unity. Unfortunately, once it is larger, the system becomes strongly coupled and the approximation breaks down.
3) Note that the nonrenormalizability of gravity is in force ('G' has units of length and is thus by powercounting nonrenormalizable) and you will find dangerous divergences for even pure gravity appearing at 2 loops. Consequently, you must truncate the series at one loop and throw out all the higher order terms (terms that involve R^3, R^4 and so forth). Thus you have a finite theory with a finite amount of couplings, that is perfectly predictive but also incomplete (technically you have a renormalization of G, a renormalization of the cosmological constant, as well as two additional couplings from new geometrical tensors).. As described, it makes no sense at very high energies (even approximately), but suffices for many intermediate regimes (such as questions about the nature of black hole radiation, simple graviton exchange diagrams, or even inflation). However, it is most assuredly not going to help you much when you get too close to a black hole singularity, or alternatively for questions about the very early universe. Right there, the full strong coupling behavior is necessary and the infinite set of couplings that you threw out (both matter and gravitational), comes back to bite you as they require an infinite amount of experiments to pin down.
Long story short, you are left with one of only two possibilities for what the tentative full high energy theory can be. Either the original theory is UV completed, where new degrees of freedom emerge, or alternatively there is a nontrivial fixed point (asymptotic safety). There is no other consistent alternative that also retains a measure of predictivity.
Now, there are a number of good arguments against even this, and in fact you can make an argument that whatever the UV completion is, it cannot be a local field theory (this arises from arguments centered around the black hole information loss paradox, see eg hep-th/0605196 for a partial review). In any event, the additional problem posed by the existence of black holes, is that we have apparently lost unitarity. Therefore the correct high energy theory must also unitarize the semiclassical physics somehow. No one has ever found a way how to do this in full generality although of course it is expected that holography is necessary (likely implying the need for stringy states)
