Stephen Fry has joined growing calls for Russia to be stripped of the Sochi 2014 Winter Olympics because of concerns over new anti-gay laws, with the actor equating the situation to the staging of the 1936 Berlin Games in Nazi Germany.

In an open letter to the International Olympic Committee and David Cameron that was also circulated to Fry's 2 million Twitter followers, he said Vladimir Putin "is making scapegoats of gay people, just as Hitler did Jews".

He added: "Putin is eerily repeating this insane crime, only this time against LGBT [lesbian, gay, bisexual, and transgender] Russians. Beatings, murders and humiliations are ignored by the police. Any defence or sane discussion of homosexuality is against the law."

The 2014 Winter Olympics, due to take place next February in the Black Sea resort and costing an estimated £35bn, are the most expensive in history and were planned as a show of strength by the Russian president.

But the growing backlash against the new laws, passed this year, has now escalated into calls for the IOC to strip Russia of the Games and, in the US, to boycott them.

Fry said: "An absolute ban on the Russian Winter Olympics of 2014 in Sochi is simply essential. Stage them elsewhere in Utah, Lillehammer, anywhere you like. At all costs, Putin cannot be seen to have the approval of the civilised world."

The new laws, passed by the Duma, prohibit "propaganda" in support of "non-traditional" sexual orientation and threaten heavy fines for anyone "promoting" homosexuality to under-18s.

They have coincided with a sharp rise in reports of homophobia and attacks on the LGBT community in Russia.

In the letter, which was also sent to the London 2012 organising committee chairman, Lord Coe, Fry added: "The summer Olympics of 2012 were one of the most glorious moments of my life and the life of my country. For there to be a Russian Winter Olympics would stain the movement forever and wipe away any of that glory."

The comedian said that despite opposing the prime minister's politics he had the "utmost respect" for Cameron for the way he had pushed gay marriage proposals through parliament and stood up for LGBT rights.

"For that I will always admire you, whatever other differences may lie between us. In the end I believe you know when a thing is wrong or right. Please act on that instinct now," he said.

A Downing Street spokesman said it had repeatedly raised the issue with Russia.

"We remain greatly concerned about the growing restrictions on LGBT freedoms in Russia and have repeatedly raised our concerns, including at the 2013 UK-Russia human rights dialogue in May. The prime minister outlined our concerns with President Putin during a meeting in Downing Street in June ahead of the G8 summit.

"We are working closely with the IOC and the BOA to ensure that the Games take place in the spirit of the Olympic Charter and are free from discrimination."

The issue has gained momentum just as Moscow prepares to host the world athletics championships. Four years after the Sochi Games, Russia is scheduled to host the football World Cup.

President Obama, who has refused to rule out a US boycott of the Games amid strained relations between the two countries over a number of issues, said he had "no patience for countries that try to treat gays or lesbians or transgender persons in ways that intimidate them or are harmful to them".

Asked about the issue by Jay Leno on his Tonight Show, Obama added: "I think they understand that for most of the countries that participate in the Olympics, we wouldn't tolerate gays and lesbians being treated differently."

Talk of a boycott recalls the 1980 summer Games in Moscow, which were boycotted by the US at the height of the cold war, and the 1984 Olympics in Los Angeles, which were boycotted by the Russians.

Last month the IOC said it would "work to ensure that the Games can take place without discrimination against athletes, officials, spectators and the media".

It said: "To that end, the IOC has received assurances from the highest level of government in Russia that the legislation will not affect those attending or taking part in the Games."

An official of Russia's main law enforcement authorities on Wednesday condemned any attempts to politicise the Games.

Konstantin Dolgov, human rights commissioner of the interior ministry, said: "We are going on the assumption that all guests and participants at the Olympics will be met with hospitality, but that at the same time they will strictly follow Russian law and will respect our country's customs and traditions."

Lawmakers had previously gone back and forth over whether Russia will enforce the gay propaganda law at the Games.

The IOC said last month it had received assurances from the Russian government that athletes and spectators would be exempt from the law, and Igor Anansky, the chairman of the state Duma committee on physical culture, sports and youth affairs, said in August the law would not be invoked during the event.

But the sports minister, Vitaly Mutko, said on 1 August the law would apply to athletes in Sochi in a statement that is likely to set the state's policy. "No one is forbidding an athlete with non-traditional sexual orientation from coming to Sochi, but if he goes on to the street and starts propagandising it, then of course he will be held accountable," Mutko said.

Russia's recent crackdown on homosexuality has seen it ban gay pride parades in Moscow and outlaw adoptions by same-sex couples and the "propaganda of non-traditional sexual orientations" among minors.

The gay propaganda law, as it is widely known, imposes fines for virtually any public discussion of LGBT issues, and foreigners face larger fines, detention for up to 15 days and deportation for violating it. Four Dutch activists who were filming a documentary about LGBT people in Murmansk were arrested under the law in July and banned from Russia for three years.

A prominent gay rights activist, Nikolai Alexeev, said although Fry's letter "won't change anything" at the Olympics, it would help raise awareness of the plight of LGBT Russians. "When respected people around the world call this out, it highlights what's going on in Russia," Alexeev said.

He said the parallels Fry draws between Russia's crackdown on LGBT rights and the start of Jewish persecution in Nazi Germany are "ever more relevant as of late", adding: "Today it's just laws, but these laws can lead to more repression and persecution of homosexual people in the future."

In March, Fry met the St Petersburg politician Vitaly Milonov, who authored a citywide ban on gay propaganda last year that helped lead to the federal ban enacted in June.

After a private discussion that was filmed for Fry's BBC2 documentary Out There, which will be screened later this year, Milonov promised to pray for Fry's soul, and the comedian referred to the meeting in his letter, describing it as an encounter with what Hannah Arendt called the "banality of evil".

On Wednesday, Russia's investigative committee questioned television host Ksenia Sobchak over accusations of slander against Duma deputy Yelena Mizulina, a sponsor of the gay propaganda law. Mizulina has asked the committee to charge Sobchak and Alexeev with slander for their criticism of her on Twitter.