Disclaimer: I know you asked for a way to gently kick someone out, but my answer is more about ways to let them play with you with reduced pain. My belief is that kicking someone out from your gaming group would damage your relationship, and there are other things to try before.
I was in a similar situation, playing a team online game together with people that are so toxic even in written chat they eventually got temporarily banned. We had some heated discussion about the game objectives and decision made, and about behavior. I kept in good terms with most of them, and set a different path with another (we met in the game so we weren't close friend though), so I believe I can share a bit my experience regarding handling the situation before, during and after the game.
If you want to play semi-competitively with a group of players (more than 2 anyway), which I can imagine at least this player wants to do, you should setup rules for how to use the vocal chat:
However if things start to look slightly bad during a game this player gets demoralised very quickly and starts criticising the other players more and more lowering the morale of the group and usually resulting in a loss.
I have a fairly simple one, that wouldn't offend anyone here: game time isn't criticism time. This is not only, from a casual gaming perspective, an undesirable thing for the whole morale and fun of the game, it is also from the perspective of a semi-competitive player an unnecessary distraction.
You can, for the need of progression, and if players agree, have this kind of retrospective criticism afterwards where people are cooler, where you have access to replay if applicable, where you can focus on the arguments and not on what's currently happening.
Try to have a discussion together before launching on agreeing with this simple rule: no criticism during the game. Would you obtain it, you would have grounding to tell live to the toxic player that he should keep his remarks for later.
If you have a loss where people want to retrospect, insist on it to be constructive criticism: blaming someone for the loss is not constructive. Loosing is part of the game, and focusing on that outcome has no improvement or team building value. Improvements opportunities are equal in won and lost games. Make sure these principles are well accepted especially by this player.
I know a few players that had personal sensitivity to anger and were not controlling themselves as well as most people do. You can first have a conversation about the possible anger issue. You could say "You seemed very angry last game. Anything wrong ? Are you still having fun playing, even when we loose ?". It's a bit fishing about whether or not they agree for an apology, and for the future whether or if they agree to countermeasures. It could be the player confess something else, like they'd like to play with someone else or to another game.
It's also about making sure they understand there is a problem, and that the subtext is that it's not really acceptable, even if that is generally useless to say. Remind them before the game, that it's just a game, and you expect it to be anger-free. You could also say, if things are not going too well you keep yourself free from cutting their microphone (if you can cut for everyone, that's best)
That seem a bit extreme, but just saying in the vocal "Sorry but I mute you for the reminder of the game" would give you have several desirable effects : you effectively sanction an undesirable behavior, you get a game free of anger, and you set an example for others to do the same. No need to add on the incident: most people would get the message fairly well.
If everything fails, you could consider playing another game with the toxic player, possibly without competitive aspect. Sometimes, it's just the game just doesn't fit the character. You could consider this as a last resort from effectively kicking him out of your gaming sessions.