Originally Posted by Arkyan

If this were necessarily true, one could make the same argument for the Earth/moon/sun system. Every time the Moon's orbit brings it directly between us and the sun, it is closest to the sun and thus experiences more gravitational pull in that direction. However as we all know, the gravitational attraction of the sun has never yanked the moon out of its orbit around the Earth. Without doing the math involved it is simple enough to say that there is a certain critical distance, after which the gravitational pull of the sun would outpull the gravity of the Earth. Anything within this point is not pulled away from the Earth because the Earth pulls harder.

There must therefore be a region of space around the moon where its gravitational pull is greater than that of the Earth. If there was not, then the rockets we fired to the moon would never have been able to land there, and would have fallen back to the Earth instead. This region of space is likely rather small, and the chances of our Moon capturing a satellite that would orbit within this critical distance are equally small, nevertheless there do exist stable orbits around the Moon (and thus any moon of any planet), its just that a configuration happening naturally is unlikely.