Reference frame, gravitation and kinetic energy English is not my primary language.  I apologize (and would like corrections) if the terms I use are not the right ones.
In a geocentric reference frame, the moon rotates around the Earth. So its kinetic energy would make it go away if not for the gravitational interaction.
Now, if I place myself in a "moon-centric" (?) reference frame, the Earth would rotate around itself, but not around the moon. So nothing would compensate for the gravitational interaction.
Of course, I am wrong somewhere in my reasoning. I would just like to know where and why.
 A: Lets distinguish 'rotation' into 'spin' (rotating around itself), and 'orbit' (orbiting around another object).  The orbital motion is due to gravity.  The 'centrifugal force' (instead of 'kinetic energy' of rotation, per se) is balanced by the 'gravitational force' [1].  
The spin of an object doesn't need an external force (gravity from another object) to balance it out.  The spin is maintained by internal forces: if we were talking about a spinning tire, the internal force would be the elasticity/cohesion of the material; in the case of the spinning earth/moon, its the internal gravity of the object upon itself.
"In a geocentric reference frame, the moon [orbits] around the Earth."  And Earth's gravity holds it there.  The moon spins very slowly, (once a month) and is kept together by its internal gravity.
In a lunocentric (?) reference frame, the earth orbits around the Moon.  And the Moon's gravity holds it there.  The earth spins quickly (once a day), and is kept together by its internal gravity.

[1]: 'Energy' and 'forces' (i.e. 'interactions') are different things, so we don't like to say they directly counteract each-other.  The 'energy' of the orbit, results from the gravitational force; but it's the centrifugal force (which is observed in a rotating reference frame) which is counteracted (balanced) by gravity.
A: As far as I understand your question, you can consider the Moon as continuously falling down to the Earth (or continuously following the moving Earth, if you want) due to the gravity. Nothing compensates gravity. However the problem is that the Moon aims not precisely into the Earth, but a bit away - that was original "error" from the Moon's first day. The Moon is just missing its target that's why it cannot collide with the Earth. Each next second the Moon understands that its route is wrong and turns a bit to corrects the way, and so happens continuously as endless cycle.
English is not my native language too, please don't take my phrases as right example. 
