Can’t say I would remain in an environment where gossip takes precedence over my own behavior in determining what kind of person I am. Boss and co-workers want to know what’s going on about me, they can come to ME. Anyone who believes lies about me told by others deserves none of my esteem. If impressing such people is my key to getting ahead, then it’s time to go somewhere else.
His problem, not yours. If he started the gossip, then of course he has no beef, if he didn’t, how he deals with it is his problem. You can still choose to ignore it.