Problem at Work

Well, I have a little problem and I am not sure what to do about. There is a 20 yr old female here that is constantly telling me how to do my job. Telling me what my boss allows and what he does not. Telling me I will never make it because I don't have the right type of education...etc. You get the picture. I see her as young and naive, still a child.
Well, on Friday she started up about how I could not do something...I tried to be nice...told her it was ok I had permission...and she kept on. And on and on and on...finally told her...It is ok, I know how to do my job. She tells me, in front of my 12 co-workers mind you, " You get an F****** attitude for nothing!" And then storms out. She Im's me...and continues the argument. I kept telling myself not to give in...but I finally did...much to my chagrin. I pretty much told her, you want an attitude I can give you an attitude. Dear Lord, when will I quit lowering myself to this childish behaviour.
Now for the problem. She has not spoken to me since. I am not mad at her...I never was...just got tired of the way she talked to me. So...I have not said anything to her. I have thought about apologizing...but then I think no....if I apologize she will think it is because I am sorry for what I said..and I am not...I don't need her telling me how to do my job. And then I thought about apologizing for getting angry...for which I am sorry and then just saying I did not appreciate the way she approached me in the situation. But if I do that...she will start arguing again. So, that is the dilemma...I don't know what to do about it. Just continue on not speaking to her since obviously she does not want to speak to me...or somehow apologize...I don't know what to do. Help me.