The petition which is hosted on the Care2 website branded Delon a "racist, homophobic misogynist".
Women and Hollywood, one of the US lobby groups at the head of the #MeToo movement in the film industry, said the decision to give him an honorary Palme d'Or "sucked".
Palme d'Or decision 'sucked'
Its founder Melissa Silverstein tweeted that she was appalled that Cannes was honouring Delon given that he "has publicly admitted to slapping women and he has claimed that being gay is against nature."
Delon, who has starred in classics such as "The Leopard", "The Swimming Pool" and "Le Samourai" admitted hitting women in November when he was questioned on French television about claims of domestic violence from his estranged son Alain-Fabien Delon.
"If a slap is macho, then I am macho," he said, adding that women had hit him and that he had never "broken" a woman.
His son said Delon had beaten his mother, breaking eight of her ribs and her nose twice.
Delon has been vocal about his sympathies towards the far-right movements in France and his interest in stopping immigration into his country.
He has also claimed publicly that gay couples shouldn't have the right to have or adopt children.
Fremaux under scrutiny
Cannes director Thierry Fremaux came to Delon's defence Monday, saying they were honouring his career not his views which the festival did not condone.
"Alain Delon is allowed to think what he likes," he told AFP.
"It's complicated to judge someone through today's lens, and judge things that were said and that happened years ago," Fremaux said, adding: "We're not giving him the Nobel Peace Prize."
In a separate development which has caused controversy and the croisette surrounds Franco-Tunisian director Abdellatif Kechiche who was selected to compete for the top prize despite police investigating him for an alleged sexual assault on a young actress. Kechiche denies the claim.
Kechiche won the Palme d'Or in 2013 with the lesbian love story "Blue Is the Warmest Colour". Its two female stars Lea Seydoux and Adele Exarchopoulos also complained about Kechiche's behaviour.
A judicial source told AFP they are still investigating a complaint by a 29-year-old actress made against him in October, although they have yet to interview the director.
Last year the Cannes festival vowed that there "must be zero tolerance with sexual harassment or abuse of any kind" amid red carpet protests led by Cate Blanchett of more than 80 Hollywood stars and women directors.