You say that there is a third party whom is deeply injured by the actions of the second party. Were you present when the injury occurred? If not, then your knowledge of the events is second-hand hearsay. Therefore, you must consider that the injured third party’s side of the story may be exaggerated because of their emotional reaction to the events, and the second party may not know that you know about what happened. There is also the chance that the third party is outright lying or manipulating you, so the second party is acting as if nothing didn’t happen, because maybe nothing really happened.
If you and the second party have never discussed the issue, you are passing judgement without allowing the person you’ve deemed guilty to defend themselves. I personally don’t like making uninformed judgements, so I do try to hear both sides of a story and attempt to glean out the truth. Usually, I find that I can relate on some levels to each of them, and that I can find fault or error in each of their actions. It is rare that such situations are so clear cut between who was “right” and who was “wrong”; life and relationships are much more complex. Getting caught in the middle of a conflict between two friends is not where I would recommend anyone to be.
If what was done was truly reprehensible (i.e. physical/sexual assault, cheating on a romantic partner, criminal behavior), then I’d simply tell the second party that I do not want to maintain a friendship with them because their actions showed that they are not trustworthy, or because they’ve shown that they don’t share the values that I consider to be important in a friend or however you want to phrase it.