[dotancohen]

It's already happened! More at the end.

This is called tidal locking, and if the universe consisted of only the Earth and moon, this would in fact happen. However, the big heavy Sun also affects both the Earth and moon's rotations.

So why do I say that it has happened? Because the moon, having significantly less mass than the Earth, is almost tidally locked to the Earth. That's why we always see the same side of the moon. So the Earth's rotation hasn't synchronized with the moon's revolution, but the moon's rotation has nearly synchronized with the Earth's revolution (actually both the Earth and moon revolve around their common barycenter).