I think that, in some situations, it's possible and not a problem to be friends with someone with whom you might otherwise have differences. I have some friends who have beliefs I disagree with, and do things I disagree with - but they're not part of our friendship because they know I have a problem with that stuff and they keep it far away from me - it doesn't enter into our friendship, and as such, our friendship is the same.
The problem is that your friend has now made his racism a part of your friendship, and you need to decide how to deal with it. I think I agree with the idea that to not say anything would be wrong - you need to object to him, and have the conversation. If it somehow resolves itself to a point where the issue comes to rest, as long as it never, ever rears its head again, and you're comfortable with maintaining your friendship, I don't see why your friendship shouldn't be able to continue.
With that said, I don't know anyone who's a racist, at least on the level being described here. Maybe some people have minor, minor prejudices based on their upbringing or whatever, but they're aware of them, don't act on them, and try to change them - and as long as they're trying to change it, I don't see how it's their fault any more than a mental disorder of some sort. I don't think I've ever met someone who's that racist, and as such, I really can't say whether it's possible to resolve issues with an unrepentant racist - I've just, to my knowledge, never met one.
What I DO disagree with is the notion being flung around that someone's being racist entirely invalidates every other good quality about them. Yes, it's a bad quality. Yes, it's stupid. Yes, it's WRONG, WRONG, WRONG, and deserving of re-education. But it doesn't invalidate their value as a person. For thousands of years, people have been racists, and still done great and good things. They were bad in their racism, but good in their other aspects, and the one does not invalidate the other. To denounce all racists as irredeemably evil monsters strikes me as hilariously, and darkly, hypocritical; you're not looking at them as people, you're looking at them as a stereotype, a statistic, and that's also wrong.
Also, I think people's talk of an inevitable level of minor racism in everyone is something of a defeatist attitude. I think it's entirely possible for people to overcome instincts - if indeed those instincts are universal, or natural, and can't be removed through early childhood education - to the point where they just don't exist anymore. It may not work for everyone, but it can work for some.