I think the “friend” who created the rumor is not really a friend; it would be very reasonable for this purported friend to confess to the young woman what really happened. If you decide to keep this friend around if he is unwilling to apologize, you’re essentially saying that you find his behavior acceptable, so don’t be surprised at a repeat offense.

As for the young woman, I’d give her plenty of distance for now. Maybe at a later date, you could offer a goodwill gesture, like a card with a sincere note (as you’ve learned, words matter).

Edit: If you were to give her a card, it would be an opportunity to put the truth into writing, explaining how the rumor got started and possibly why. No need to write a novel. Just keep it simple and sweet, like “I told my stupid friend I thought you were cute” or “I told my former friend that I thought you were cute” (if the dimwit is truly gone, as that would speak to your character). No references to potential relationship, love or sex is a necessity, because you’ll scare her off or creep her out. Also, don’t bother to tell her how nice you are; this is best demonstrated via behavior. I think focusing on slowly building a friendship first is likely the only way you’ll regain any trust.

Btw if you decide to do any kind gesture for the young woman but she sees you still hanging out with the twit, it’s pointless (unless he has explained and apologized profusely to the young woman and she has accepted).