The South was already well on its way to becoming solidly Republican, and with the exception of Louisiana and Texas, it really couldn't go back. We might have stronger Dixiecrat showings in 68 and maybe 72 but all the Republicans have to do is stay morally conservative and the South will come. Blacks started going Democrat after WWII but it was gradual.
There would be no 'black vote' here. If Republicans aren't seen as racists then, as being the most socially concervative of all, they would mostly be Republican in the South although the North will probably be split.
There won't be a Clinton/Gore as they won't be elected in the first place (or not as Democrats). But there might not be a Bush II either as Texas would have remained Democrat for a while longer.
(let me explain: Southen Democrats have always been America's unofficial third party, they're just as conservative as Republicans, and only slightly more liberal economically. They didn't become Republicans until lately because of coordination problems as they didn't want to lose their importance in washington. Thats particually strong in Texas/Louisiana, Texas had democrat governors until the mid-90's and Louisiana hasn't had a Republican governor since reconstruction. I'm republican and very partisan but it didn't bother me that much when Blanco-D was elected governor of LA, her platfor and Bobby Jindal's were almost identical.)