To set up the integrals you need the potentials. It seems to me that you have to have some equation for the potentials before you can do the integrals. Am I right or wrong? Experts?
If you treat general relativity (GR) as a field theory, trying to quantize it using path integrals, the best you can do is to describe the quantization of *linearized* GR, meaning you look at small fluctuations about a smooth background spacetime (described by a metric, either flat or curved...let's just stick with the Minkowski metric). Then if you carry out the path integral quantization, and get past all the subtleties involved, you find that the theory is ultimately non-renormalizable. By our interpretation and criterion of renormalizability for a "physical theory", we conclude that such an attempt (to describe quantum general relativity as a field theory on some smooth, classical, spacetime) is wrong.
If you try to go back and get a "Schrodinger equation" for quantum gravity, describing the evolution of the metric you reach whole new problems that are too detailed to go into here. The problem is that we didn't have a background independent quantum mechanics not too long ago. However, in the past two decades, Abhay Ashtekar has been greatly responsible for constructing such a quantum theory of gravity (though the story is far from over and it is not nearly a final model of quantum gravity). Gravitation is not a simply theory, even classical GR has thorny issues, let alone the quantum theory!