Moreover, the Spanish Inquisition acted on the power of royal decrees, so they very much enforced laws of the state, which means your example falls flat even on its most basic premise.
You seem to either misunderstand or misrepresent the point of the protests against Brendan Eich. Mozilla employees weren't trying to abridge his rights to free speech or to vote according to his beliefs, they simply didn't want to follow a person actively working towards making their lives measurably worse for no good reason. Of course getting him to resign doesn't make him change his voting; that was never the point! Brendan Eich isn't a child, and people's main objective isn't to discipline him and make him understand what he did wrong- it's to limit his damage! Why allow someone to infect an organization you care about and depend on with their toxic beliefs? Again, a CEO's biases don't amount to a hurtful code comment; they are steering the entire company culture!
I agree with you that the leadership does set the culture and tone for the company. But I also believe that forcing leaders to transmute their personal lives into the moral symbol of an amorphous corporate entity is somewhat inhumane. Do we have to only have leaders who lead pious, uncontroversial personal lives? Isn't there a balance to strike between this person's personal and work life?
Famously, Steve Jobs was an asshole. Jeff Bezos is another asshole. There's undoubtedly been other asshole leaders of companies. As leaders, they set their company's cultures. But where is the moral outrage asking them to step down from causing harm to their employees? You read about them in many books and blogs - terrified employees, managers making bad decisions out of fear, difficult work/life balance. And yet, if you look at the actual moral outrage from the tech world, this is business as usual. But it's harmful! On the scales of moral justice, it would seem terrorizing your workforce can be okay, depending on the form it takes.
This is another stupid, poor comparison on my part, but I'm trying (in vain probably) to get at this: what is considered "beyond the pale" of harm probably is not based on some kind of measurable outcome, but rather what gets people more angry - specifically the anger rooted in moral exclusion. In addition I'm saying that it's possible that keeping people around, regardless of their questionable personal ethics, might not be a cultural catastrophe.
I've worked with many people over the years, some of them in positions of leadership with controversial views. But the company culture encouraged open and honest communication, and its strength helped people have the occasional difficult conversation in an open way without it becoming a toxic working environment. That's certainly not always possible. But it was proof enough to me that by sticking by these people, we could eventually explain and help open up the thoughts of someone who would go on to lead others.
Another argument could be made that even following someone who's making one's life worse may have, in the grand scheme, a greater-good impact. One example might be Mozilla fighting for Internet... whatever it fights for, and another might be the power of the supply chain to provide cheaper goods (Amazon, Wal-Mart) or an alternative platform for creative work (Apple). So there may be reasons (excuses?) for following a leader who is making your life hell. You'd have to ask those employees why they do it. I do it because one coal mine is as good as the next. I suppose if you want to keep the most coal miners doing the most work you need a mine boss that inspires them, so a mine boss that nobody likes and doesn't inspire great work is probably worth chucking. But if the workers can put the boss out of their mind and still get their work done, who really cares about the mine boss? At what point can/should we stop focusing on individual morality and instead focus on practicality? Maybe that's a dangerous direction in itself.
Eich had a choice to save himself: renounce his views. Of course the Jews were not given a choice during the Holocaust. But as a different example, they were given a choice to renounce their faith during the Spanish Inquisition.
Clearly it's ridiculous for me to compare being burned alive to being corporately run out on a rail. Yet the choices offered, from a moral standpoint, were similar. Choose to renounce who you are and what you've done and you can stay a good member of society; stand your moral ground and be persecuted. In either case, the crime was not violating the Law of the State, but the moral law of a certain majority of society.
How do we judge if we are being treated justly? Is the law always just? Are actions outside the scope of law always just? In truth, this is often subject to the time and place. Even today, we sometimes treat people unjustly within the scope of the law. That's why I suggest ethics that do not hang it's treatment of people on the power to act from personal morality alone.
I think rather than say "repent or be shunned", other tactics could lead to changing the subject's mind or actions. Or even accepting that exiling the person does not change much materially about the world. They'll still be the same person in exile, doing the same things, so what was the benefit personally or to society of removing them from the group?
If the "harm done" was merely to the emotional peace of the part of the group that has to come to terms with the morality of people acting within the law, this seems like a reason to keep the person. Because again, even if working outside Mozilla, Eich may vote the same, so the "harm" from a legal standpoint against gay marriage is unmoved either way. The only effect of the backlash was purely to the emotional peace of either (and to the overall quality of life of Eich). Keeping them could at least allow a rapport to form and possibly change views, on either side, without causing further harm. But this is just one case, so this may not work in other examples.