Sure, our employee is at-will–but can we fire him just because he might have a dark side?

Q: We had an employee call out for one day recently because he had been arrested on a domestic violence charge. He did not violate any attendance policy, and we're not sure if or when there will be a trial. This man has long been rumored in the office to be an abuser, and the police have been called to his home on several occasions. Though he's an at-will employee, can we realistically fire him if he's broken none of our rules? – Paul, Florida