I feel bad for him, it is only because of his public position that this is an issue. If you dig deep enough, none of us should keep our jobs.
There's a difference. A CEO leads a company. He or she is the face of the company. Whether they like it or not, if their actions and beliefs will reflect on the company. Holding unpopular views or views that are contrary to those of the company or, at the very least, contrary to the rest of board will make it likely that you'll be "encouraged to explore other opportunities".
Businesses operate in their own best interests. They are not democracies. Seeing the negative publicity and the potential repercussions to their image/reputation, the rest of the Mozilla board felt that having him as CEO would be bad for business.
Freedom of speech does not guarantee freedom of consequences. Publicly taking an unpopular socially conservative position while holding a position of power in a business catering to a fairly socially liberal base is going to generate some consequences. It would be like a Republican Senator in Kansas growing a brain and giving a speech lauding the research and science behind evolution, climate change, vaccines, etc. You wouldn't expect that senator to get re-elected and wouldn't be surprised if a recall vote came screaming out of Fox News.