A motivation to develop the Bohr model of the atom was according to Wikipedia:
The laws of classical mechanics (i.e. the Larmor formula), predict that the electron will release electromagnetic radiation while orbiting a nucleus. Because the electron would lose energy, it would rapidly spiral inwards, collapsing into the nucleus on a timescale of around 16 picoseconds. This atom model is disastrous, because it predicts that all atoms are unstable.
To overcome this difficulty Bohr had to postulate that electrons don't radiate in orbits where the angular momentum of the electron is a integer multiple of Plancks constant.
Now lets assume we had an atom of neutral particles holding together only by gravity. If a small light particle orbits a much more heavy one, I believe the light particle would follow a Schwarzschild geodesic. But neutral particles following a Schwarzschild geodesic don't seem to loose energy by radiating gravitational waves. So I think, if we had atoms holding together only by gravity, there would be no need to introduce Bohrs postulates and there would be no need to quantize the angular momentum of the orbits.
Is this really true? Would "gravitational atoms" be stable without quantization?