As I predicted, he went the classic route of the political loser (Trent Lott being a recent example that comes to mind):
(1) Initially refuse, publicly and repeatedly, to apologise. Insist you have nothing to apologise for, and that it is in fact your critics who ought to be ashamed ("my opponents have sunk to a new low" was how Howard put it).
(2) After things start to snowball, issue a late, very grudging, and obviously insincere pseudo-apology in the form of some variation of "Some people were too stupid or sensitive to get what I meant, and I regret their pinhead feelings got hurt."
(This is how far Dean's gotten so far--it remains to be seen if he'll continue to #3)
(3) After things have gotten way out of hand, and you look like political roadkill, issue an abject apology (or many of them) with no qualifications--but no appearance of sincerity either, since it is clearly issued as a desperate effort to save one's own political skin.
(4) Go down the tubes, one way or another.
Some candidates are too prideful to go for #3, or maybe they know #3 never works anyway, and just go straight to #4 (getting the nomination isn't enough to disprove #4, btw--only beating Bush would do so definitively). That'd be my prediction for Dean, as the "regret" he expressed Wednesday was, as Morton Kondrake noted to general laughter, "as abject an apology as you'll ever get from Dean."