The pop star accused the Amazon Prime motoring show - hosted by Jeremy Clarkson, James May and Richard Hammond - of making "repulsive" jokes at the expense of the LGBT community.
During the latest episode of the show, the trio drive through Colombia, with Clarkson in a Jeep. His co-hosts suggests the vehicle is for gay people, with Hammond suggesting Clarkson needs a new grooming routine and a change of clothes.
He said: "Maybe some nice chaps. Suede but ventilated at the back." Later, The Weather Girls' hit It's Raining Men plays from the Jeep's stereo.
After raising concerns over the episode on Twitter, openly gay Young said on Tuesday evening he had received no response from Amazon, so industry regulator Ofcom "beckons".
He added: "Enough is enough and I'm pissed and fed up. I want Amazonprime and the producers of grand tour to meet young lgbt who want to kill themselves because of shaming and laughter and normalising of shaming homophobic narratives.
"your time is up and it's time to take a stand. Is there a legal case ? Who knows . ....... watch this space."
Young, who rose to fame after winning Pop Idol, said the show was guilty of stereotyping.
He said: "I DON'T drive a Wrangler Jeep. I DON'T wear pink shirts . I DON'T wear arseless chaps. You can be honest and funny without this ridiculous 'lad' ooh being gay and let's laugh about it mentality . It's repulsive and how DARE you do it and put it out @PrimeVideo."
Clarkson, who used to present the BBC's Top Gear alongside Hammond and May, addressed the controversy on Twitter.
Replying to someone complaining who said their "homophobic jokes have to stop", he said: "What homophobic jokes?"
Amazon Prime has been contacted for comment.
Clarkson was dismissed from the BBC in 2015 following a "fracas" with a producer, while his two co-hosts followed him voluntarily.
This is not the first time Clarkson, Hammond and May have been accused of homophobia.
In a 2016 episode of The Grand Tour, Hammond was condemned for saying he does not eat ice cream because he is "straight".
Clarkson was criticised in 2006 after calling a Daihatsu Copen "a bit gay" on an episode of Top Gear.