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The job of a CEO is to lead the company. If his/her personal views are affecting the public image of the company, the use of the company's products, and the effectiveness of the company's employees, then that is a problem for the company.
If it makes business sense for Mozilla to have a different CEO, then they should appoint a different CEO. If what Mr. Eich brings outweighs any negative effects (incurred and expected) wrt. any other potential CEOs, then, by all means, keep him.

But I don't believe that there are no other candidates who are also well-qualified, and who would not kick up this kind of reaction.