Should I tell my manager if a coworker is making untrue allegations about them behind their back?

I’m going to try and ask while giving just enough details not to doxx myself..

I like my manager. We’re not friends, but she’s fair and knows what she’s doing, and we’ve never had any problems in the 3 years I’ve worked for her. One of my colleagues really hates her, though, which is fine, because everyone is allowed to have their own opinions or whatever. However, recently this colleague has started over sharing a lot with me (I think he may like me, but I have never given him any reason to think it would be reciprocated. I’m just generally friendly with everyone). On several occasions now he has stepped past normal venting about work, and accused our manager of some things that I know for a fact aren’t true. For example, he claimed that she didn’t give a high-profile project to another one of our colleagues “because he’s black”, but I know for a fact that our colleague didn’t want that project and asked not to be put on it for other reasons. He also made an off-hand statement about how it was “weird that the only gay person on our team has a smaller desk than everyone else”, but again I know the reason for this (from talking to said gay woman) that she specifically asked for that desk last year because it’s closest to a window. There have been several examples, and it’s always him making a comment that insinuates she’s discriminating against a protected class.

Each time he has said things like this, I’ve told him why he’s wrong, and the last time he did it I asked him to stop making those kinds of accusations because they were making me uncomfortable. But now I’m wondering - should I tell my boss? I don’t want to be a “tattle tale” and stir up trouble, but he seems to be trying to build some kind of discrimination case against her. These things are easily verified as untrue, so I’m not too worried about what would happen if he went to HR or a lawyer, but I’m worried that if I somehow get pulled into things that I’ll be asked why I didn’t come forward and say something. As a manager would you want to know, and would you keep it in confidence if someone came and told you this?

TLDR - a coworker is gossiping about my manager including multiple untrue allegations of discrimination. Should I tell her?