Growing up with that kind of publicity, and his parents' connections, he had little choice but to go into show business. In 1965, he started a boy band with his friends Dean Paul Martin (Dean Martin's son) and Billy Hinsche.
Here's Lucy (1968-72), as Lucy's teenage son Craig. And he landed roles in lots of movies.
Others were gay-subtext movies about the tragic consequences of boys who fall in love with boys (Billy Two Hats, Joyride, To Kill a Cop).
In 1983, he starred in a buddy-bonding series, Automan, about a computer whiz (Desi) who creates a computerized superhero (the hunky Chuck Wagner). Unfortunately, it was promoted as a comedy, and Desi didn't do comedy. He thought it would be a gritty urban drama.
After 13 episodes he left, and hasn't done a lot of acting since. He is still involved with his music, and he owns the Boulder Theater in Boulder City, Nevada.
Although he was the subject of many gay rumors during the 1970s (my friend Cesar claimed to have hooked up with him), Desi was linked to several women, including Liza Minnelli, and he was married to Amy Arnaz from 1987 to her death in 2015.
See: Cesar Tells about his Hookup with Desi Arnaz Jr.