An object ONLY revolves around its C of G. If you say that the moon hangs towards us because of gravity, then you are saying that it is rotating around a point eccentric to the C of G, thus letting the C of G come closer to the earth. This is impossible.

To add to an earlier point, it is sufficient to treat planetary gravity as being r distance from the planet's centre of gravity rather than the planets surface. (Although this does not apply where r is less than the planet's radius.)