Eesh...yes, somewhat awkward and difficult.
I agree with hbelden; you have to deal with these things as a stage manager first. Last year I worked on a show where pretty much all of the cast and I and the director were friends (small ensemble show of only 6 actors); but one actor and the director in particular were particularly good friends -- one day in rehearsal she gave him some constructive criticism on a beat, and on break he took my outside and said that he felt really attacked by her comment and that it was out of line and unfair of her, and could I please talk to her about it? I said no -- she's the director, she's directing you, and if you don't agree with what she's saying, then bring it up in rehearsal and discuss it actively and constructively. I reassured him that nothing she said was directed as a personal attack and also reassured him that if a director ever did make a personal attack on an actor, then I would talk with them about it. He was mad at me for the rest of the rehearsal day, but the next day when he had calmed down, called to tell me that he was thankful I took the "bitchy SM stand" rather than the "friendly oh-its-okay-side."
What was the extent of the "rudeness"? Was it personal comments, or was it just crude remarks in general that the cast member took offense at? I agree with hbelden; you can't take sides -- you have be the neutral presence and do whatever is possible to diffuse the situation.
I don't think you were wrong in telling her honestly what it would mean and how people would perceive things if she quit, and of course she is hurting because you dealt with her as the stage manager of the show rather than as an objective friend -- but the thing is, in this case, you're not an objective friend; and anything pertaining to the show you have to deal with as the stage manager -- even if its at midnight or the wee hours of the morning.