I would have a serious performance conversation with the manager and explain to them that their job as a manager is to take important but sometimes difficult decisions. If they can't bring themselves to do this, they shouldn't be a manager.
The outcome you want is the shittiness to stop and you have an objective transcript of everything that happened in slack, so assuming the shittiness is real this isn't even remotely hard.
Option 1: If the behaviour isn't that bad or is in some sense borderline or open to interpretation, you speak to them in person, say the thing that happened that needs to stop, state clearly why it isn't in line with your values, is damaging to the team/whatever the problem is, and state your expectation that it will stop and if it doesn't immediately stop it will affect their future with the firm. You follow up with an email recapping all of the conversation (in particular the statement that if it happens again they're out) and copy HR. You need to do option 1 if you have been allowing this behaviour to slide in the past. It's important to do this with all parties who have been indulging in this behaviour and don't show any kind of favouritism.
Option 2: If it's not the first time (ie you've already done some version of option 1 and it hasn't stopped) or the behaviour is really egregious you book a 1:1 with the person and when they show up HR is there with you. You show them the thing on slack that is happening that isn't ok and any previous "option 1" recap emails. You ask them to explain any extenuating circumstances. You terminate them there and then unless they have a really compelling story. You send an email to the HR person thanking them for handling the situation (especially since the reason you got here is that you have made mistakes as a manager), summarising the conversation and your thought process that meant option 1 was not appropriate. If they've done this before you have the email track from when you did option 1 before to add to their evidence trail.
For at-will employees there's no way any kind of wrongful termination will stick if you follow this kind of fair and rigorous process, you document conversations you have had and your rationale for dismissal is reasonable. For employees in a European or UK employment context, you need to be more aware of the surrounding context and make sure they have actually breached your policies and then follow your disciplinary process appropriately. If you work with your HR colleagues this isn't hard.
[edit] There is also an "Option 0" above which is less heavy-handed. Part of being a good manager is knowing when to take the person out for a coffee, put a hand on their shoulder and just give them the "Come on you know this isn't ok" conversation. Then you follow up with an email (but don't copy HR) and make sure you save it in case you need to escalate further. Just make sure you don't option 0 your buddies and escalate everyone else because that is seriously bad.