Existing answers are fine if both of you were employees, but since the other person is a consultant and not an employee of your company, things are a bit different.
First of all, you didn't handle this wrong per se, but you didn't give this person the benefit of the doubt (not sure if this was intentional or not). You could have asked your manager or any colleagues on the call (if any) for their opinion, to clear the possibility that you misread the situation. Instead, you outright filed both a formal complaint (email to your manager) and an informal one over non-standard channels (you telling this person's boss).
HR is never going to get involved in this, because this contractor isn't managed by HR. They can't get fired from your company because they don't work there, they can't get reprimanded by your HR department, and they don't need to follow any of your HR department's rules or your company's formal rules or informal rules (culture). They are an employee of a different company. That's important to remember.
For that reason, the correct escalation path is to talk to your manager, and let your manager resolve or escalate accordingly.
It's important that you understand that what you did was escalate this. We're in no position to tell you whether that was the correct move or not, since you didn't describe the incident in detail. However, now that you did escalate it, the best thing you can do is let your manager handle it (which is literally what you asked them to do, when you sent that email).
If this happens again, be sure to document it appropriately (recording of the call if they're available, detailed account from you, etc), and discuss it with your manager. Do not attempt to resolve this on your own without discussing with your manager. That would have been ideal at the beginning (depending on how bad it was, and we don't know that), but that ship has sailed now.
Basically, act like you would if this was another employee of your company and you had reported this to HR.
Just so you know, I've seen multiple times contracts with either service providers or even clients get terminated because one person mistreated company employees and their boss didn't do anything to change that. It's usually because of repeated incidents, and it takes a while to process since it does impact your company's operations and potentially revenue, but it can happen.