If someone is expressing political or even religious views that are so extreme that they create a hostile work environment, I don't see an issue with firing them over it.
(I am not interested in debating whether or not Damore's views reached the "hostile work environment" threshold)
There is a process, and step one in that process is asking them to please stop sharing their views in the workplace.
Expressing extremely wrong views can't be the standard for creating a hostile work environment because people are wrong about a bunch of stuff and the only way to figure that out is to express the view and be told why it is wrong.
Particularly in the Damore case, the standard shouldn't be how a third party feels. There is a long history of seeing how that standard plays out, it'll end up hurting minority voices. The standard needs to be how a first party responds when asked to do something.
Damore would have had to prove that there was a active discrimination at google toward some political views. He had some good evidence for that (in terms of internal conversations and witnesses) but it is not so clear whether that was enough, or even if it was necessarily illegal in this case.
The causes of action in the lawsuit were unlawful termination on account of political activities (which are protected in California) and sex (which is protected federally). Google made three motions to dismiss and lost all of them. Does that mean Google is guilty? Of course not. All it means is that the claim stated by Damore, if true, could convince a reasonable jury that Google violated his civil rights.
Google would much rather pay out settlement money than risk having an unflattering jury verdict or have to release even more unflattering documents in discovery. I'm sure Damore would rather not go to trial if he could help it as well.
People say "political views", that's how he might frame it, but I would assume Google would frame it as they fired him for making statements that women are less capable.
Edit: burn, karma, burn
Some facts are not open to interpretation.