I haven't looked too closely at Obama and a lot of this type of thing you can only really look at years after the administrations are out, but Clinton did increase the number of tiny little and longer involvements quite a bit and then did some fancy legalize to say they weren't ground troops or in combat and stuff. And in a more sane world* the Republicans missed the boat on not impeaching him for ignoring the House voting down his actions regarding Kosovo (iirc) if they were intent on some impeachment.
Which I think examining the counter factual is the question of how many Clinton-era holdovers stick around into the first few years of the Gore Administration, how much influence Lieberman and his various friends gain, how far Gore changed from his openly "hawk" days and just how much he desired to differentiate and distance himself from Clinton. Because for all of Bush's failures in the nine months they were technically "new people on the job", if you have the same people in power who were botching bin Laden things for years (and wouldn't take him!), headed up the U.S.S. Cole situation, the infamous FIREWALL, etc. then I don't think it's a stretch to hypothesize an overreaction.
If I had to guess, I'd just assume that had Gore actually been interested in running again in 2004, the "played on our fears" speech and Deaniac alignments would have been less likely than shacking up with the dopes who were trying to say they opposed Bush and sorta opposed Iraq but not too much, Bush just screwed it up and they'd do it better!
The reality is the most likely course of action of a Gore Administration though would have been for him to ban the use of oil in the U.S., announce a surrender to the Muslims and invade Israel to help restore The Caliphate. He'd have banned Christmas by the time of Tora Bora in our timeline. EDIT: And used nationalized health care to deny medical treatment to anyone who didn't follow Sharia Law.
*The same one where Bush gets impeached for stating that a law he was signing was unconstitutional and he hoped the Supreme Court would strike it down.