AZ Does not require a license or permit at all, so long as you lawfully possess the gun.
There have been many answers suggested, but none, I think, better than the OP's actual response. Well done. "Have a good day" and walking away left her nothing on which to hang her hat, so to speak. A smart-ass remark, a "crazy" ("the lettuce can HEAR us!") response, or a loud vocal one such as "STOP TOUCHING ME" would only add fuel to her mistaken impression of us. Imagine the post she'd have made over on Daily Kos or Huffington Post about confronting the big, bad gun owner. With this as described, she can spin it to say that he just had no answer at all or she can lie and say he "sputtered off" or even that he loudly berated her (hey, I said she was lying...), but she could not escape the fact that he proved himself a level-headed individual. She just failed to admit to herself that she already knew that, else she'd never have confronted him.
@IamaClone45: "I'm not with the police department": Might I suggest, "I'm not with the local police, no Ma'am." with a little smile. If you need to elaborate for some reason, "I'm responsible for the safety of someone very important." Of course, that person is yourself. I think that tends to legitimize her apparent belief that only those employed to provide protection should be armed.
I also like the response, "You are welcome to your opinion", but that is in no way better than just "Have a good day." Again, OP: Well done.
Blessings,
Bill
Allow me to suggest a modification to "You are welcome to your opinion" ?
"You are welcome to your opinion thanks to the 1st Amendment, just as I am welcome to carry this firearm under the 2nd. Have a good day!"