But ministers are coming under mounting pressure to drop him over his previous comments on LGBTQ rights, women and climate change.
The letter, which has also been signed by former Doctor Who boss Russell T Davies and activist Lord Cashman, says: “This is a man who described himself as ‘threatened by homosexuality’, and vigorously campaigned against the ultimately successful referendum in Australia to allow same-sex couples to marry.
“This man is not fit to be representing the UK as our trade envoy. If the government is truly committed to an outward-looking future for Britain, to tackling climate change, and to creating an equal society for all, it should reconsider its proposed appointment of Tony Abbott.”
Others signatories include the founders of Pride in London and the CEO of leading HIV charity, the Terrence Higgins Trust.
It came as former Australian PM Julia Gillard said she stood by “every word” of a speech she made in 2012 in which she berated Mr Abbott for alleged sexism.
She told Sky News: “I certainly stand by every word of that speech. I doubt that I’ll ever be able to put it better in my life than the things that I said that day on misogyny and so I’ll let that speech speak for itself.”
In the original speech she gave to the Australian parliament, she said: “If he wants to know what misogyny looks like in modern Australia, he doesn’t need a motion in the House of Representatives, he needs a mirror.”
Labour leader Sir Keir Starmer said he has “real concerns” about Mr Abbott and would not appoint him if he was Prime Minister.
Former Tory health secretary Jeremy Hunt said he would be “surprised” if Mr Abbott was made a trade envoy, while Tory former minister Caroline Nokes said it was an “awful” idea and called him a “misogynist”
Health Secretary Matt Hancock yesterday defended Mr Abbott as “an expert in trade” despite the allegations of misogyny and homophobia.
Transport Secretary Grant Shapps dodged questions about Mr Abbott this morning, telling Sky News: “I hate to bring us back to reality but he hasn’t been appointed to anything and as far as I know there haven’t been any appointments made.
“There are lots of people with whom their comments I vehemently disagree but I’m not into defending people who are not actually carrying out any role for the British Government.”
Mr Abbott fought against the introduction of same-sex marriage in Australia and has also suggested that climate change is “probably doing good”.
He reportedly said that men are more adapted to “exercise authority or to issue command” and that abortion is “the easy way out”.
Defending himself, Mr Abbott said: “Inevitably if you have convictions, you’ll draw criticism. But if you want to get things done you need people with convictions.”
The Prime Minister’s official spokesman has insisted that “no decision” on the board’s make-up had been made and has declined to comment on the “political debate” surrounding Mr Abbott.
Mr Abbott only served as leader in Australia for two years before he was ousted by rivals inside his own party.
His sister Christine Forster has tweeted to back her brother, saying: “It is nothing short of dishonesty for commentators and politicians who do not know Tony to label him a ‘homophobe and a misogynist’ for the purposes of scoring cheap political points.
“As a woman who has always been part of his life and who came out to him as gay in my early 40s, I know incontrovertibly that Tony is neither of those things.”