My bff did something like this to me once and now I’m very careful not to tell her anything that might be mistakenly divulged. It was easy to forgive considering it was unintentional and it wasn’t all that bad, but it took awhile to feel comfortable around her again. However, I’m not in love with her lol, so I don’t need to place my complete trust in her the same way as I would my partner. One thing you do need to do is give him space; he’s processing. If you become too clingy or pushy it will make him even more uncomfortable. It’s a good sign that he isn’t treating you coldly because if that were the case then I would say that it’s done. Just give it some time and it will fade, more so when the family issue is resolved. It doesn’t hurt to check up on him every once in awhile, but don’t push him. Whether you lose the relationship really depends on how big that indiscretion is, but I don’t think you’ll lose the friendship.