Do you think he has a "right" to be paid to spread his gangrenous hatred opinion on a television program?
Well, to be fair, he didn't say anything like that on the show. I don't think A&E would want to stray into that territory on that show.
One could make an argument that his private personal views shouldn't matter to the show if said views are never expressed on the show. However, one can also argue that the show is what is giving him a platform outside of the show (GQ wouldn't have interviewed him if he wasn't on the show) and that A&E has every right to say "we don't want to give him that platform".
It's overall an interesting issue, and a reflection of the way the country is moving morally. If he had supported gay rights, and A&E fired him for that even though the show itself never discusses the topic, the roles here would be reversed. Gay rights advocates would be complaining and the opposite side would be saying "A&E has every right to do that".
Regardless of the issue, a private company like A&E has every right to decide that it doesn't want to be associated with a certain view point. People have every right to complain about or support A&E's actions. Eventually one side will have the upper hand. The fact that A&E see's Robertson's views as a liability is a good indication as to the direction the country is headed on the issue (i.e. the right direction).
His views are increasingly seen as unacceptable, much like if had started spouting off about how blacks are inferior to whites. Stuff like that makes you a social outcast.
ETA: He probably would have been OK if he had been ore tactful in his response. If he said something more along the lines of "I'm a religious man, and part of my religious beliefs is that homosexuality is immoral. I don't treat homosexual people different than anyone else, but I just don't see it as morally acceptable. That's all I'm gonna say about that." It would have been somewhat an issue, but it would have probably just blown over. But he took it too far.