Should private beliefs be enough to prevent someone from heading a project they helped found?
No, but he didn't keep his beliefs private, he tried to turn them into law. And that still doesn't mean he can't head the project, it just means a lot of people may walk away from it, and Mozilla.org needs to consider that.
Is the backlash itself justified? Well, to some, including myself, it is a bit like supporting segregation in the 1950s. Right now, it is a mainstream political view to believe that gay people should not have equal rights. There's a hundred thousand years of evolution behind that belief, and it is not realistic to expect everyone to switch that internal belief off at the drop of a hat -- no more than it was possible for people in the 1950s to instantly accept equal treatment of black people.
But what good people did do in the 1950s was stop expressing their prejudice. They stopped supporting segregation, and stopped saying that they found it to be an acceptable practice. Most of them still had that deep internal programming. Most people still have it to some extent today. Hundreds of thousands of years of "different looking means dangerous" genetic programming isn't going to go away overnight. But we have reached a point where we treat those beliefs as flawed baser instincts, like the desire to hit a person over the head and steal their BMW. We repress those feelings because we believe in being better than that.
We have reached a point in our society where prejudicial treatment of black people is no longer accepted. We will reach that point with gay people too, and Mozilla will be as embarrassed of having an unrepentant bigot for a CEO as Walt Disney Corp is of Walt's anti-Semitism. It is not that Mozilla should be forbidden from doing so, it is just a question of showing good judgment.
Mozilla, tell Eich to figure it out and recant his position. It's OK to be unable to overcome your baser instincts; that is a reality of being a flawed human. I'll admit that my instinctive reaction to the idea of gay sex is not pleasant. But it is not OK to express prejudicial beliefs or to support prejudicial laws.