It's an interesting question and reframe you've posed and one that I'll get slammed for responding to I reckon, but I've never let that stop me before. I think you've made an unfair comparison (but a good one) and I'll try my best to argue my way out of it.
If your example was that a man threatened to bring his friends down to rape the woman... that wouldn't constitute assault in the same way as your example suggests. He could argue 'heat of the moment' and 'she approached me' and a whole host of other things - but we've been given a window into the messy misogyny beneath the surface. That's not the same as committing an act of assault, albeit unpleasant.
Amy Cooper bared her racist fangs, but I think where we disagree is whether calling the police constitutes an act of assault. I don't think it does, but then I'm not from the USA and I'm a Penguin. I think people have the right to hold unpleasant prejudiced and nasty thoughts and views about the world - provided they don't transgress the legal framework in which those views exist. That's liberalism. When we begin subjectively policing what is or isn't morally right and enforcing our views we'll be dragged into a legal grey area that it'll be difficult to extricate ourselves from. Would it matter if Christian Cooper had eight previous convictions for sexual assault? No. And yes. Depends how you look at it.
So, with my reputation for being a controversial bird and a half. No, I don't think he should be fired, even in the example you gave. I can see why the women in the office might have a problem with that (rightly so) - and I think they absolutely lobby for him never to spend any time alone with any of them. But it might be that he's never had any issues with any women at work. It might be that he's had issues with all of them. People have complicated lives and complicated multifaceted versions of the self.
But I don't think it's the place of a company to make judgments about things that happen externally from their premises or sphere of work. I think we have a framework for how people deal with interpersonal conflict in wider society and it's the law/police/courts. People's personal lives and their working lives should be kept separate wherever possible. I would put in a employers clause which says 'if you are found guilty of a crime (not a tort) we reserve the right to terminate your employment at our discretion' - that's a different matter from firing someone prima facie because of something on social media. I think as long as the definition of an 'ist' and a 'phobe' remains up for grabs, we must be very careful what we allow to happen. I think in this instance your 'ist' label is fair and even handed, but I can't help but think that 'ist' label in the hands of less balanced people will lead to absolute chaos.
You're absolutely right, we agree in principle. We just disagree about where the line needs to be and who gets to draw it. If you're interested in something tangential to this discussion but which plays into the same sort of problems, you might enjoy this little bit of British legal chicanery.