Fire the individual has as about as much sense, as kicking someone out of an airport because they happen to have a bottle with more than 3.4 ounce of liquid. Sure, there might be a rule or code, but there is also a thing called overreaction.
I bet he won't make the same mistake again. Perhaps we're all so used to going through the 6 month HR-slap-on-the-wrist process that we have lost touch of what it actually takes to change a person's behavior.
What mistake is it that this person should have changed?
Not to use any from of possible objectionable joke/language when talking privately with a friend but in a public setting? I guess he should never again say anything that has to do with politics, religion, sex, violence, or technical standards.
Or is it to avoid any places with people he doesn't know. Avoid all conferences, bars, cities, towns, buss stations and so on? Only ever be at places where he know he can speak freely without anyone taking "offense" if what he said is overheard.
Firing seems like an extreme overreaction at the very least, although if this were part of a pattern of behavior, or if the guy were otherwise marginal, I could understand it (even in cases where I wouldn't agree with it). More severe action might also be warranted if the employer were government or some other special cases where even the hint of sexism were a problem (e.g. a rape crisis center employee or something) -- but not a shitty game monetization company.
Reputationally, the company has a right to be both disappointed and concerned.
I don't really support Adria's approach to this (especially how she now appears to be dismissing even fair criticism as "trolling"). But the company has a reasonable argument for dismissal, whether this had splashed across the net or not.