Majority rules. If they've been asked to step down as a result of a vote between the others and their is reaction is to throw a fit, it doesn't sound like they're a good fit for staff. The best thing to do would be to would be for staff to explain their feelings about the situation and that it's not personal. Hopefully differences can be set aside but if that person is going to be bitter, it's not staff's job to coddle them.

If a fit was not thrown and that person requested another shot, maybe a compromise can be reached with a 30 day probationary period or something, depending on the circumstances. No need to be heartless if they're willing to pull through with some effort.