I think it is really a question of a) would it be legal to fire him and b) would it be a larger negative impact to the company to have him there or not. That's really a decision only you can make.
As long as it is legal to fire him, I'd say separate yourself from the personal issue as much as you can while trying to evaluate the overall impact of the situation. Part of that evaluation is taking a meaningful look at how it has impacted your working relationship, but also looking at what it would cost the company to lose him vs the problems the interpersonal trouble would cause. Only you can really be the judge of that equation.