Don't worry (too much), I think this happens to everybody at some point in their academic life. Your case isn't even particularly egregious, given that the new faculty member just started as an assistant professor, so likely she actually was a grad student not very long ago. I once idly asked a fellow conference attendant who she is working with, and she told me that she is a tenured associate professor. Embarrassing? Certainly, but not the end of the world. You apologize and move on.
The problem is that the person who I was introduced to is female, and I am afraid that I might have offended her.
I am not sure whether being female makes her more or less likely to be offended. However, I can't help but notice that most such stories are about female professors, so it seems to me that we jump to conclusions about them more freely than for males (or we guys just have trouble assessing women's age accurately). In any way, I have learned my lesson to be extra-careful before assuming that a young-looking female academic is a student, and you should too.
Am I just being paranoid or will some people really feel offended from such a comment?
I would not worry too much. You can bring it up lightly on a good ocassion and apologize, if such an ocassion presents itself in the next weeks. However, don't make it a bigger deal than it is, and if she seems unbothered or hasn't even noticed, then let it go.