If I wasn't close to her I probably wouldn't bother. but if she was someone whose friendship I valued, I'd probably say something along the lines of "I realize that what I said yesterday upset you. That wasn't my intention and I'm sorry if I hurt your feelings." I am a little confused though, because you said you didn't mean to offend her, but then you say they have been winding you up, and that you're tired of being a pushover, which sort of implies that you *did* mean to say whatever it was that offended her. If you didn't mean to upset her, or if she wasn't part of the group that's been winding you up, then an apology doesn't make you a pushover. We all say things that accidentally offend another at some point and it's an honest mistake.
However, if it was a response to something she's been doing to upset you, then you could still find a middle ground between giving her an apology/keeping the peace and also standing up for yourself. You could apologize, but also say something like "I have to be honest though, part of it was in response to X which has been happening over the past few weeks. I didn't say anything back then, but it's been bothering me and I just wanted you to know."