The show hosted by filmmaker Mr Karan Johar in December comically insulted several Bollywood actors in front of a large star-strewn audience in a format common in Western countries but rarely seen in India.
A video of the show, called AIB Knockout which is heavy with profanities and sexual references, attracted more than eight million hits in just a few days when it was uploaded on Youtube late last month.
But the show sparked a fierce backlash, with religious groups urging police to file charges against the producers, and the right-wing Maharashtra government, of which Mumbai is the capital, ordering an investigation.
The producers said they were pulling the video for "pragmatic" reasons, but added that India needed to confront the fact that freedom of expression was being curtailed.
"There's a larger cultural conversation going on here, where we're at the very edges of what it's okay to say," the Mumbai-based comedy collective, All India Bakchod, said in a statement.
"And it's a conversation we need to have now because the world we live in is entirely too complicated to be run by silences.
"We still stand by our belief in the right to absolute freedom of expression for us and for anyone who has anything to say about anything at all," the statement added.
India has a history of official and self-censorship, with British author Salman Rushdie's 1988 book "The Satanic Verses" banned for allegedly insulting Islam.
Acclaimed Tamil-language author Mr PerumalMurugan announced last month he was quitting writing following protests by right-wing Hindu and caste groups over one of his books.
The "roast" sparked a storm on social media this week, with a member of India's powerful censor board slamming it on Twitter as "a porn show on stage".
The show includes Johar and Bollywood actors Arjun Kapoor and Ranveer Singh singing a reworking of a song with the Hindi word for penis substituting some lyrics.
But hundreds of supporters including Bollywood actors and producers also hit back, urging the right to freedom of artistic expression.
"I hope our humour gets sharper, our dissent more rigorous, our satire more offensive — and till we arrive there, we stand by AIB Knockout!" independent filmmaker Anand Gandhi tweeted.
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