How do I greet a non-Muslim female; if she reaches her hand out gesturing hand-shake...how do I not offend her?

You cant really not do anything not to offend her, just shake hands, nothing bad will happen.

I really, really agree with the answer that says put your hands together and bow. It's not a deep bow, it is just an acknowledgement with your head. Then have direct eye contact and give a smile. That will keep the lady from being offended and it will let her know that it is your custom not to touch. As for those who said get a real religion, they are rude, and if other people would be so respectful to ladies there may not be as many teen pregnancies and single moms in western countries.

You should shake her hand it's alright =) Actually I bet a lot of women would consider it rude if you don't do it when she holds out her hand to you and you shouldn't want to make a bad first impression.

well i know why your asking the question, Political correctness and the fact she may be offended, so i agree with a lot of the comments just nod and say Hi.

just shake her hand. u dont have to hug her. western society is about touching or being physical so its rude if u dont at least shake her hand.

Put your hands together and bow as a greeting.

Pretend like you have to sneeze, achooo in your hand that way she won't want to shake it.

Or convert to a real religion that doesn't require such nonsense.