Eich's beliefs mean nothing
No, but his actions of agression, based on those beliefs, towards people who didn't do him any harm, kind of speak against his ability to lead.
And yes, if an employee of mine made public comments about not wanting me as CEO because of my politics, I would show him the door in an instant.
Good, do that, but remember that you are also just an employee and can get the boot just as well for firing people just because they don't like you or agree with you. If you can't take criticism, you will be either fired or left to rule over cowards.
Nothing is easier than to denounce the evildoer; nothing is more difficult than to understand him. - Fyodor Dostoevski