Yes, if someone launches an Internet lynch mob of social justice warriors at me that threatens the business, of course I would be fired.
No company wants to be perceived as "the place that employs the most-hated sexist in the world" if they depend on selling to regular people. An Internet lynch mob can ruin SE rankings, launch DDoS attacks, etc. Anything that attracts one is a liability.
That is of course a very stupid behaviour of such company - by letting someone go because of SE rankings or DDoS attacks of angry, misguided crowd, you're just showing that this is a great way to influence you. So the next time the misguided crowd wants you to fire someone? They'll find something to get themselves offended about, get angry and DDoS you.
Companies are inherently shortsighted and beholden to their customers and shareholders (in public companies). Prolonged social attacks against a company may drive customers away or drop stock value, both of which negatively impact the company (and more importantly management) far more than letting one employee go. This probably significantly harms employee morale, but I doubt that's their primary concern.
I know the reasoning, but I suspect it's based on wrong assumptions. Management is afraid of taking a stand and doing the right thing, even though it would most likely end up well for the company, not bad. I see this as risk aversion.
It would never have reached the proportions it did if it wasn't for all the people "defending" the guy and then keep bringing it up time and time again to make an argument.
