Nick Vallelonga tweeted his support for Donald Trump, who said he had seen "thousands and thousands" cheering - a claim that's been widely discredited.
Vallelonga wrote that Mr Trump was "100% correct", but has now said sorry "for the hurt I have caused".
He won a Golden Globe Award on Sunday, and the Oscars are now six weeks away.
Trump 'wrong' on 9/11 cheering claim
Vallelonga wrote Green Book about the story of his father Tony Lip, who drove black concert pianist Don Shirley through the US Deep South amid the racism of the 1960s.
Viggo Mortensen plays the driver, while Mahershala Ali, a Muslim, portrays Shirley. But its Oscar chances may be affected by a string of recent controversies.
Shirley's family have criticised the film, saying it is filled with "lies"; co-writer and director Peter Farrelly has apologised after an old interview surfaced in which he talked about flashing his genitals; and Mortensen has said sorry for using the N word in an interview.
The latest controversy comes after the re-emergence of the tweet Vallelonga sent following a rally in November 2015, where Mr Trump - then a presidential candidate - said: "I watched in Jersey City, New Jersey, where thousands and thousands of people were cheering as that building was coming down."
Vallelonga wrote: "100% correct. Muslims in Jersey City cheering when towers went down. I saw it, as you did, possibly on local CBS news."
Image copyrightREUTERSImage captionNick Vallelonga (left), Viggo Mortensen (centre) and Brian Currie attended the National Board of Review Awards
There has never been any evidence to support Mr Trump's claim that "thousands" celebrated after the attacks in 2001.
A clip has emerged of a WCBS correspondent at the time reporting that police found eight men celebrating in one building in New Jersey, but it did not show footage of the men.
In a statement reported by Hollywood Reporter, Variety and other sites, Vallelonga said: "I want to apologise. I spent my life trying to bring this story of overcoming differences and finding common ground to the screen, and I am incredibly sorry to everyone associated with Green Book.
'This lesson is not lost on me'
"I especially deeply apologise to the brilliant and kind Mahershala Ali, and all members of the Muslim faith, for the hurt I have caused. I am also sorry to my late father who changed so much from Dr Shirley's friendship and I promise this lesson is not lost on me.
"Green Book is a story about love, acceptance and overcoming barriers, and I will do better."
Participant Media, which produced the film, also released a statement saying: "We find Mr Vallelonga's Twitter post offensive, dangerous, and antithetical to Participant Media's values. We reject it in no uncertain terms."
La La Land producer Jordan Horowitz was among those criticising the writer on Twitter, calling his comments "too disgusting".
Nick Vallelonga wrote Green Book. My industry just gave him a Golden Globe for writing. This remains on his timeline.
Mahershala Ali is a Muslim, and a beautiful, generous and kind man.
This is all just too disgusting.
Given the Shirley family disowning the film, Viggo using the N-word, the Peter Farrelly story, and now Nick Vallelonga being anti-Muslim, I've become more flustered that GREEN BOOK is taking attention from more deserving films.
On Sunday, Vallelonga picked up the Golden Globe for best screenplay alongside Farrelly and co-writer Brian Currie. The Golden Globes are among the key precursors to the Oscars.
The controversies have come at a crucial time in the Oscars race - voting for the nominees opened on 7 January and closes on 14 January. The ceremony itself will take place on 24 February.
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