your question is about the analogy between the mathematical form of potentials Coulomb and Newton, but physical situations are very different.
Yes, you could do it, use wave functions on the Hydrogen atom for solve your problem, but where is the sense?
You can do it if you consider Earth and Moon like material points, but their magnitude is very bigger than their quantum wave lenght.
Of course, you can try to enjoy yourself applying quantum mechanics for problem like this, but in this situation you must consider magnitude of Earth and Moon...so your problem is very difficult. And the wave lenghts in this problem make you consider the particles (electrons, protons, neutrons,...) that constitute Earth and Moon too...But thi is like use Relativistical Theory for calculate the motion when you walk on the road...An unuseful and terrible complication ;-)