Yes, he likely would've. The president on 9/11 would most likely be reelected in 2004. His opponent might be John McCain, since McCain was considered pretty hawkish and credible on foreign policy, was a war hero and did decently the 2000 GOP primaries.
Below is a possible 2004 map with Gore and Liebermann as the Democratic candidates and McCain and Lindsey Graham. (The reason McCain might pick Graham is because he, Liebermann and Graham were close friends, all three foreign policy hawks. It would also be interesting to see a race featuring all three of them.)
