Not in my experience, and it's not the law (as taught in the myriad of training courses I've endured over the years). The man should have an opportunity to learn from this mistake, then be fired when he proves incapable of learning.
Providing the warning and training is enough to legally cover the company's ass if the guy screws up again.
Yes, it's a big mistake to make, but when your hiring pool is predominately college graduates, it's pretty likely they don't know the law (or how to navigate romance in the workplace) either. Time to educate them.
EDIT: It's also important to get the other side of the story; why did this guy feel it was appropriate to approach the woman in the first place? Were there conversations not included in the record? Was there body language the man misinterpreted?