You will never get a white person to apologize for saying/doing something racist. Apologizing would mean admitting that I am a racist! And, we've all decided that being a racist is one of the worst things you can be (I think it falls just behind child rapist). So, sorry black people, no apologies for you!
Well I think the problem there is "Racism" is such a broad term. It runs the gamut from George Wallace blocking the door the the University of Alabama to Black People Love Us. There's no line drawn between oldstyle "Blacks are subhuman" racism and "stepping on toes because you're ignorant of black culture" racism. No one white person in their right mind these days wants to risk being lumped with the Klan.
I also recently watched Do the Right Thing and one thing that really struck me this time was that every time Sal got into a conflict with Buggin' out and started taking things too far, like reaching for the baseball bat or making obscene gestures in front of children, it was his blatantly racist son Pino that restrained him. I don't know if it was intentional or not, but that struck me as a point that it's the people who keep their racism hidden, even from themselves, who are the ones to really watch out for.