YTA. Your wife must be a great wife if the worst thing you could think of to fit in with your buddies when they were complaining about their SOs was that she had a perfectly reasonable boundary about her private medical information. "God, my wife won't let me betray her trust about a private issue that affects no one but her, such a burden."

Honestly you're TA for the old and tired "gotta hate the ol' ball and chain" routine alone before you compounded it with betraying your wife's trust and her sensitive medical issue. They may have been doctors, but they weren't her doctors. Your TA level only increased by telling her she was overreacting and overdramatic.
These aren't random friends of yours she won't see ever (though her reaction still would have been appropriate even if they were), these are people in her workplace, that had the gall to approach her about it even knowing she didn't want anyone to know.
For as long as she works there, she'll know they know. She'll wonder how many more people they've told. If her own spouse can't keep their mouth shut, why should she trust her spouse's friends to do so? This isn't one instance of people finding out. This is a potentially ongoing stressful situation that she will never escape unless she quits her job.

I think a little repeated recrimination was more than warranted.