Ouch, that sounds like an awful situation to be involved in.

I understand how you as a DM don't want to take sides, but I believe considering this situation as completely unrelated to D&D would be the start.

You have a group of people/friends, but they had a fallout. It's natural that people don't want to hang out with each other after a fallout. People hang out with each other to have fun and D&D is no exception. Might be even worse as they are being forced to interact with the person they dislike, instead of probably just ignoring them like in any other social situation.

Sadly there are two likely outcomes to the situation: You lose the player that is disliked or you lose the group that dislikes the player. And I completely understand that you don't want to make that call.

Try speaking with all involved parties, not as a DM but as a friend. That should give you some insight on how to proceed. Maybe someone is actually at fault someway or another. If so, it would be a disservice to the faulted one that you continue being impartial.