From Alison Green writing at New York magazine:

I have a conundrum. A co-worker of mine has said some pretty awful things on social media, and I don’t know what, if anything, to do about it.

I work with this person occasionally. When I was new and had to ask him a few questions about a project, I googled him to check out his LinkedIn and so on, and the first thing that came up was his Twitter feed, with the company name in his bio. I added him to a list and went on with my day. In the past few months, though, his feed has turned into a fat-shaming Milo Yiannopoulos–loving bro-fest. He removed the company’s name a few weeks ago — now it has something about “triggering supplicating millennial babies.” Ohhhkay. One of his most recent tweets crossed a major line for me: He used a homophobic slur, specifically to taunt people for being “too sensitive” online.

…

Most of the time I’m a pretty staunch advocate that what you do in your private life is your business. But that’s really only true as long as you’re not hurting anyone else. If you’re publicly spewing hateful, vile things about other people — and you’re doing it in this era’s equivalent of the town square — it’s fair game for an employer to have concerns about that.