JLMACN, it all depends on your current situation;
(a) does your homie really like this girl? (b) does this girl really like your homie? (c) do you think they'll be together for a long time?
If they're not happy together and your homie just wants her for sex, then definitely don't apologise. If, however, the answer to these questions is yes, then you should apologise. If they're potentially going to be together for a long time, then it's best you both settle your differences, and by the sounds of it, you have to make the first peace offering. I'd tell her something like 'my homie really likes you, and I know he doesn't like us at each others throats - I'm sorry for what I've done and I'd like us to get along, for his sake'.
She seems like the decision maker in the relationship, and if they're together for long enough, he'll end up going with her over you - it always happens. Don't give her the chance - if your homie knows that you've tried your hardest to be diplomatic and nice to her and that she's the one fucking with you, he'll side with you when shit hits the fan and kick that bitch to the curb.