Legal Question in Business Law in California

(a) you can sue, but (b) your likelihood of success is very small. Men simply don't have a protectible legal interest in girlfriends. She might have a better chance with a lawsuit based upon some kind of annoyance or stalking-type claim

A man has the right to ask, just as a woman has the right to say no. You haven't provided any facts to support any viable legal claim.

If your girlfriend feels harassed, she may have viable claims against the salesman. You don't, whether she feels harassed or not.

Automobile dealerships are regulated by the state DMV. The behavior you describe sounds unethical, and you should make a complaint with the DMV about the dealership, which is responsible for the actions of its employee. The DMV is a bit slow about responding, but they will eventually follow up.