What happens if a jewish man kissed another woman while married and that man feels extremely guilty and vows never to do it ever again. What does he do? Just teshuva? Does he have to tell his jewish wife?

Shlom, Thank you for your question. You are correct in feeling that this is an extremely grave sin, that certainly needs teshuvah. I cannot stress the seriousness of being totally (that means 100%) faithful to one’s spouse. There is no excuse, no justification, at all for a husband (or wife) to have any physical or romantic connection outside of their marriage. I know that in the modern western culture this is not a given (and perhaps the opposite is true) – but I can say with complete confidence – the total fidelity to ones marriage is of prime importance in Judaism. The process of teshuvah involves regret, confessing in prayer to Hashem, and asking His forgiveness, and a resolve not to sin again. In order for this to take hold, may I suggest adding extra stringencies in the area of modesty in order to make sure that he does not come anywhere near this sin again. Perhaps a good place to start are the laws of negiah (forbidden physical contact between men and women) and yichud (the laws surrounding being alone with women), If he is careful with these laws, they will serve him well as a fence around further sin. As to telling his wife – that is a very delicate question. In general, it may lead to bigger problems in the marriage, and is probably not a good idea. However, ideally, if the couple has a very good strong relationship, and telling the wife will not destroy the marriage – then, yes, honesty is a good thing in marriage. So, without knowing the couple, it is hard to give a clear answer. Blessings.