SCOTTSDALE, Ariz.—Despite middling results with the New York Jets, Rex Ryan became one of the most beloved coaches in the National Football League, winning followers with his sincerity, salty mouth and unapologetic bravado.

Jets players and fans adored him. He returned the sentiment, most indelibly by getting a tattoo on his upper right arm that depicted his wife, Michelle, as a pinup model wearing nothing but a Jets jersey.

In...