Actually, you do not have a right to freedom from verbal attacks at work. Be very careful that you do not jeopardize your own job, your attendance, etc by being "so upset that I broke down and left" because even if someone cussed you out or screamed and threatened you at work that does not mean you had the right to leave the work site without authorization.
If you are actually assaulted physically at work, call the police and press charges. You did the right thing by coming first to your direct supervisor, and then informing your regional manager of this situation. But now, talk to them again, tell them clearly and firmly that if you are actually assaulted by this person, you will hold them responsible and that you will press charges. You can ask them to change your work assignment or their work assignment so that you no longer have to work with this person who yelled at you if you want to. But really, they don't have to do that. If they want to, they could fire you because they want to, or because there's a problem between you and the other person, or just about for any reason.
As cbg has mentioned, depending a lot on the specific circumstances, but unless there is an EEOC related aspect to this situation, you can't file a grievance anywhere or take this to the "labor board" or sue them or anything, so there is no next step except to try to resolve the problem with your employer. If they won't work with you, you have the right to quit.