I’m sorry. [ I don’t remember the incident / remember the incident that way ]. Regardless, it [ upset / hurt ] you, and I apologize for that. I’m not sure I understand what I did that [ hurt / upset ] you, but I am working to understand and to be a better person.
Again, I apologize for the hurt I’ve caused you, regardless of my intent.
This is simple. It’s to the point. It acknowledges the hurt and pain you caused – whether you remember it, intended it, or not. And then you move on from there.
Don’t be a sarcastic jerk, only to find out you were factually wrong.
Be like Bill.