I can see why Republicans want to get rid of Mark Sanford: he's an embarrassment. But I can't see why Democrats would have any objection to what he did. After all, it was just about sex. (And love, and marriage, and faithfulness, and some other things, but nothing about government.)
I can see, however, why Sanford would not resign. A better question would be, why would he? If he resigns, he admits that he screwed up in a big way, and his political career is over. If he doesn't, maybe his career is over, but maybe it isn't. Already his story has been pushed off the front page by Sarah Palin's resignation, and who knows what else will happen between now and 2010 or 2012. People have a way of forgetting things. I don't mean anyone will forget his affair entirely, but its importance will surely diminish over time. By 2012, Republicans might be begging Sanford to run for president. They might not, I admit; I'm just saying it is a possibility. If he finishes out his term honourably, he could go out reasonably popular; and if there are major problems, such as the recession continuing for several years, people may start to regard his governorship as a golden age and his affair a trivial matter compared with urgent matters of state. In either case, what does he gain by stepping down? He has already done all the damage to his marriage that he possibly could, and his political career is going nowhere if he resigns.