Aegon never bothered with people insulting him, but there was a limit. You wouldn't intentionally kill an ant, but if it bites you, then matters become different.
Gabriel was like that ant for him.
He didn't care about her insulting him, as it seemed to be her personality. But if he let her go without doing anything, then she would think he was a pushover.
Especially since this was being broadcasted. Once people saw that he could be stepped on and walked over, they would gain confidence and do it themselves.
House Augustus and House Valeria had way too many enemies, so him showing weakness was something they had been waiting for a long time.
Best to solve it before it became a problem.
'Sorry, old man. But I don't care if she is your granddaughter.'
