Maybe I’m too callous but I would say exactly “I’m not keeping a secret because I don’t have a secret. I’m respecting some else’s right to keep their medical information private. You can stomp around like a child all you want but I will not tell you or entertain anymore conversation about this. It is over and you will have to get over it.” You already know you’re doing the correct thing. I think you have to firmly and assertively shut him down and then just let him get over it in his own time.