This is a totally hypothetical situation that would never really happen. My wife is angry at me. She thinks she told me to pick her up from the train station, and I never showed up. She waited in the rain for half an hour, couldn't find a taxi and in the end had to call a friend because my phone was off. I am absolutely sure she did not ask me to pick her up. Now she won't speak to me until I give her an apology. Can I apologize when I don't believe I did anything wrong?
I don't usually answer such far-fetched made-up scenarios. But this time I will make an exception.
When we say, "I am sorry," we are making a statement about ourselves - I am remorseful, I regret my actions, and I hope not to repeat them.
But an apology is not just about you and your feelings. It is about the person you hurt. You don't apologize just to absolve yourself from guilt, but more to acknowledge that you are the cause of someone else's pain, and take responsibility for it.
This means that even if you are completely in the right and really did nothing wrong, even if the other person misinterpreted your words or actions, even if you did nothing to regret, nevertheless if someone else is hurting you need to apologize for that.
Only G-d knows who is right and who is wrong in your case. But we all know who is hurting. So you need to be big enough to apologize.
But beware. If you say, "I am sorry for any pain you felt," it will come across as empty words, as if she has a problem and you feel sorry for her. You need to arouse true feelings of empathy for her, and real regret for your part in what happened.
Just imagine her standing in the rain, drenched and dejected, calling your unresponsive phone and not knowing what to do. And in her mind, you caused all this. You owe her a sincere apology. Hypothetically.