the President (as well as congress) have the final say whether or not we engage in war or pull out. take a look at somalia... highly visual issues happened (downed helicopters, 18 soldiers killed), and clinton pulled us out. i think it was a huge mistake to get involved, only then to run away when the whole mission doesn't go 100% our way. something like that requires complete dedication to see it all the way through. not that i blame clinton wholly, as congress would've voted us to pull out regardless of what clinton said.
but no, bush doesn't call all the shots. he leaves that to the military leadrs. and of course it really does matter who we voted in, because gore would've chosen a completely different cabinet than bush did.