The situation is no fault of your own but if your anger continues to lead to depression because the trust and friendship you have developed has been eroded, then it's YOU who is the most important person in this issue. You need to take the steps to ensure that you are 'safe' to further your well being and therefore your career.
As Manalone says, you're really not at fault here for what happened. I don't know if it was right to spill the beans about your closeted friend to your other friend, but he was really being left in the dark so I guess you had no other choice. After all, you didn't want misinformation to harm your friendship with the other doctor yet.
I don't know why this guy reacted the way he did, and stopped talking to you and your other friend because of your rejection. People have different ways of handling things. It's understandable he might be a little hurt, but at the same time he is a married man with children. And it's a little wrong for him to automatically think that because he revealed his sexuality and his interest in you both at the same time, that you might say yes. And then what? He divorces his wife and runs away with you? He knew there was a possibility that you would not reciprocate those feelings for him, or even be interested in him in that way just because you are also gay. I understand he probably feels embarrassed for opening up about it and having things turn out that way, but it's not like you were going to out him and embarrass him - you just wanted to put the conversation behind you and continue being friends. He has put a wedge in your friendship.
It's possible that maybe he isn't lying when he tells other people the reason why you don't talk is because you don't open up. Maybe there are other reasons why you aren't talking besides that conversation? You know, I think about things I said about my ex to people as we were breaking up, and after we broke up. All of it was truthful, it was how I was feeling about her and how she acted at the time. Is it an accurate portrait of the entire situation? Is it all-encompassing of the feelings I felt and the things that happened during our long and drawn-out break-up? Not at all. But it was part of it. So, maybe part of the reason why he stopped talking is because that is one of the things that caused a rift between you two, besides his botched reveal.
Or, maybe it is just what he told people to get them off of his case. Maybe he didn't mean to single you out as the source of the problem, but when discussing how you two no longer talk, the only substitution for what really happened that made sense is to say that you don't open up. When you think about it, it's really not that insulting. So you might not open up much, so what? Yeah he should have come up with a better excuse if that's what he did, but at least he didn't come out and make up some huge lie that makes you look like a terrible person.
Whatever he is telling people, that is just how he is dealing with things.
Obviously your friendship with these two people means a lot to you. Is the other doctor at least still talking to you? I mean, you could try to make an effort to extend an olive branch to this guy, even though it's not really your fault that the two of you aren't talking. Maybe he is having trouble coming to terms with his sexuality and has nobody else to talk about it with? Again, it's really not your job to be the person he has to do that with, but you could offer a friendly ear if he is willing to talk about it AND not view you as more than a friend.
At least you are on a rotating through different specialties. This means, you won't directly cross paths with this other doctor forever. Your rotation will take you to other places as you continue your training. (Surely this other doctor would have realized this, that you might not really be working together forever - and that you might want someone a little closer to your own age.) I would just focus on your work, and on worthwhile friendships. If this guy isn't talking to you, maybe at least you won't have to worry about him causing trouble for you - maybe he is just moving on from things. If you are concerned that he might do something to retaliate against you, I would see about transferring or something.
I suggest that you be more private with your personal life in your professional life.
You are going to run into these Drs again some time; you want to leave everyone with the very best impression possible.
This man may have felt that he was being tested or invited into your personal life. Now - embarrassed, self-shamed and surprised - he has shunned you. He may even be afraid you will hold this over his head, professionally.
I hope this is a learning experience for all.
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