It depends a bit on your workplace and how big it is. If it's a small business where there's only 4 or 5 of you, I'd probably be making sure my CV was up-to-date, because you won't be able to avoid him. But I am assuming you work for a bigger organisation.
I would have said start with your boss, as she's already aware of what he's said about you and her, and given that he lied about you, I'd be far more inclined to believe her than him. But you're not sure if she's telling the truth, which suggests you don't fully trust her, so it maybe better not to talk to her. Can you talk to her manager?
Go to HR. Before you go write down what you remember about what he's said and when, and anyone else involved - you, your boss, other team members, whoever. This helps get it straight in your head as much as anything else. It may count as harassment, and it's definitely causing disruption to the team, and upsetting you as an individual, which could mean your productivity is affected, and therefore, there's a financial effect, too. (This always counts for more at our place, anyway.
Also, whatever you decide to do, keep notes of what happens when - date, time, who, summary of what's been said and any decisions/actions. Keep any relevant mails, too. He's already lied about what's happened and tried to make out he's the victim - you should be fully prepared for that to happen again, particularly if he feels he's been told off.
He sounds like an immature, odious git, and should be stamped on before he harasses anyone else in the workplace.