OK, you win. Let's say honor and generosity of spirit are worthless and should have been cast off ages ago. Here are my suggestions for effectively addressing the situation while not apologizing*.
The lawyerly approach:
A dispassionate and disinterested review of the facts will confirm that I have not engaged in any misconduct. I did my best to navigate through the challenge we faced, and I deeply regret our miscommunication.
The underused if:
If... I misunderstood you in any way, I deeply regret it. (It is crucial to emphasize the if.)
The existential approach:
Mistakes were made, but by whom? Well, it doesn't matter. Nothing is real.
The apology for something else approach:
Hey, man, that whole f*cking thing was just totally f*cked up. Real sh*t. Pardon my French. Sorry, I got carried away.
The OJ Simpson approach:
I didn't make a mistake. But, if I had made a mistake, this is how I would have done it. (Give a near fictional account of your behavior.)
The innocent-by-association approach:
I think a lot of folks misunderstood that issue. We're all human.
The I'm wonderful approach:
I understand how we might have disagreed. I am very understanding of human nature, and have years of experience being a decent human being. I actually learned a lot about disagreements while volunteering at a soup-kitchen every month for the last 7 years. Disagreements were quite common in the food line.
The Happy Meal approach:
About that thing: I apolog... Oh! What have we here? This is fun!
The reverse-psychology approach:
Disagreement is healthy. We should enjoy a good laugh about this. Laughter is healthy, too. I'm so glad we can agree.
*In no particular order of innocence.