What is the theory concerning gravitational atoms? Imagine sometime in the distance future where the expansion of the Universe has created vast areas of empty space, light years wide.
Imagine, say, a neutron and a neutrino bound together as a gravitationaly bound atom. The "atom" in it's lowest energy state would be several light years across such is the weakness of the gravitational force.
Presumably the atom could jump to a higher energy state by absorbing a graviton. 
Zooming out of this picture so that a light year looks only a few cm wide, would this look in any way like a normal atom but just on a vastly different scale?
Could these gravitational atoms form molecules?
Are the rules of these quantum gravitational systems known? Or does one need some unknown quantum theory of gravity?
In the distant future could most of the matter be made up of these gravitationaly bound atoms?
 A: You don't want a neutron and a neutrino, because neutrons are unstable. You could do this with, say, a carbon-12 atom and a neutrino, or with two neutrinos.
Quantum gravity is not needed for this kind of thing, e.g., people have done neutron interferometry in a gravitational field and have gotten the results you expect based on ordinary quantum mechanics. Since the force is $1/r^2$, the system would behave exactly like a hydrogen atom in terms of its wavefunctions and energy levels. The energy and distance scales would just be different.
For more complicated systems, the structure would not be the same as for ordinary matter, since there would be no electrical repulsions, only attraction.
A: Ben Crowell's answer is of course right, but let me add a few things:
I see mainly three problems:

*

*Without the electric charge there would be no molecules (in your case), since there are no valence particles, and in your case the covalent bond would not work because there would be no particle that could create a common field around two cores (in your case neutrons).
Even if there would be a particle that tried to create a common field (by orbiting two cores, or two neutrons), that particle would not be able to stay in orbit because of the relative motion of the two cores (unless in the case maybe where the two cores would not move relative to each other).


*In our micro world, atoms have a EM field around them, and not just particles orbiting, as per QM, electrons exist everywhere in the field, and they are not classically orbiting.


*In our micro world, the very thing that withstands gravity's crushing force, is the EM repulsion of the atoms' Em fields. That is what gives matter stiffness, and volume, that is why matter is not crushing into a small volume by gravity's force (except black holes). Without that repulsion, in your case, two cores (two neutrons) would anyway come together by gravity's force. And in this case there could be no molecules, with volume at least.
