Suppose Joe Leiberman had won the DEM nomination in 2004, would Bush still be re-elected?
Hard to say. On the one had, Lieberman would be very attractive to parts of the electorate that leaned Republican in that election, but were otherwise moderate or Democratically inclined.
On the other hand, Lieberman would be viscerally repulsive to a number of core Democratic constituencies and special interest blocs. Senator Lieberman was a quintessential moderate, and unlike Senator Kerry, Clinton, etc was largely unapologetic about his vote(s) and support for the Iraq war authorization and subsequent supplemental funds. It's very, very dangerous for an American presidential candidate to have only soft support of their party's base (See McCain, John; 2008) for a great number of reasons.
Senator Lieberman's candidacy would essentially be a very difficult balancing act. He may capture large numbers of centrists and even Republicans, at the great risk of Democrats defecting to a third party candidate (Angry, rebuffed Howard Dean leads the Democratic wing of the Democratic party! ...Or something.) or simply just staying home on election day. However, attempting to woo the left-wing of the Democratic coalition could gravely undermine Lieberman's moderate reputation, and potentially demonstrate serious contradictions with his record of conduct in the Senate.
On the whole, assuming that Lieberman somehow managed to grab the nomination, he ought to have had at least as good a chance as John Kerry, even with the unique set of disadvantages he possesses.
The elephant in the room (or donkey, in this case) is that Senator Lieberman didn't have a very compelling means by which to acquire the nomination in the first place. I suppose it's just barely possible that if the race breaks down to Howard Dean and Joe Lieberman, the latter could pull off victories in enough primaries with the support of independent voters in the relevant states. That would be a very tough and enervating primary battle though, and probably a bitter one.