You have to work out how important this issue is to you - is it a deal breaker if he won't apologise? If not, can't you just agree to differ? Don't let your pride ruin an otherwise good relationship.

I agree. As hard as it is for you to see, he probably thinks you're in the wrong and you're having a hard time accepting it. If it's a big thing, you need to talk about it. If it's just a little tiff, let it go.