It sounds like you and your dad are close. Is it possible that this is the first time that he realized that your and his world views differed on something that he considered important? Maybe he was processing that.that's exactly what i was thinking - i remember when i had to gently 'educate' my father when he couldn't understand why i didn't agree with him on something. and i remember when i realized that my own DS was ... well , a person, with his *own* views and morals.
I don't think either of you handled it badly.
This is what I'm thinking.
And the idea that when people are upset (and your dad *is* allowed to be upset with you, and to express that), they need time to get over being mad. And that one should give them that time.
So perhaps the better thing to do would have been for Dad to excuse himself and find something to do in another room--go sharpen the lawnmower blades?--while he deals with the strong emotions.
Or, since he didn't, maybe your move would have been for -you- to leave the room--go wander over and help Mom fold laundry.
Neither of you did those things, so while perhaps you neither one were deeply wise, I don't think either of you were required to be. It would be hard to be! You're emotionally involved; it takes some detachment (hey, I can backseat drive w/ the best of them, LOL!).
Your dad hurt your feelings; you revealed that. (And I note that your mom didn't think you were wrong to want to leave.) I think that's fine too. Your dad deserves to see the evidence of how his treatment of you affects you.
Going forward, I would say--don't really revisit the issue, not the original dispute nor the apology sequence. Leave well enough alone. Least said, soonest mended.
And, is it not written, "It won't get better if you pick at it"?
And be gentle with one another for a little while, so you can reinforce the love and consideration that is the core of your relationship