A couple reasons. Firstly, Aerys was holding Elia as a hostage, so Doran withdrawing his support for the crown means his sister dies. Secondly, regardless of any insult, Rhaegar is still the crown prince and Aegon, Doran’s nephew, is his heir. Doran’s going to fight for his nephew’s rights even if he thinks his brother-in-law is an adulterous asshole.