First off, moving the moon is not a matter of overcoming gravity, it is a matter of overcoming inertia, still an impressive feat. However, as someone correctly pointed out, this was done with "flight strength," not muscle power.
Superman (as far as I know) did not defeat Doomsday with flight-strength but with muscle power, ie, fisticuffs.
Putting the asteroid incident aside for the moment, Hulk has not moved any celestial bodies because given that he doesn't have any "flight strength," he can't be put in the position of doing so, not without some really GAB PIS. This does not necessarily mean, however, that if given some kind of cosmic leverage/bracing, the Hulk still couldn't do it even in theory. Given what the Hulk has (supposidly) been able to do (ie, that asteroid thingie), he would appear to have the potential to match Superman's feats (provided he gets mad enough), using not flight-strength but muscle power, which, again, is not what Superman used to move the moon. This of course then begs the question: using just muscle power, could Superman still have moved the moon?
DD vs Hulk could go either way, provided on how the fight is set up. You wanna talk arm-wrestling, weight-lifting or some other pure-strength feat--then Hulk wins.