Christian Institute was speaking after Mr Cameron called legalising gay marriage in 2013 one of the "great steps" he had taken as PM in his resignation speech after Britain voted to leave the European Union by 52% to 48%.
Mr Cameron originally called the referendum and backed the Remain campaign.
He will now lead the Cabinet until a new party leader and Prime Minister is selected before the October Conservative Party conference.
Announcing his resignation (below) to the world he said: "I believe we've made great steps, with more people in work than ever before in our history, with reforms to welfare and education, increasing people's life chances, building a bigger and stronger society, keeping our promises to the poorest people in the world and enabling those who love each other to get married whatever their sexuality, but above all restoring Britain's economic strength."
Less than half of David Cameron's party voted in favour of same-sex marriage, with the Prime Minister needing support from other parties to get the legislation through.
While hundreds of thousands of people signed petitions urging it to remain illegal, many of them Christians, there was also strong support for gay marriage across the UK.
The Christian Institute's Deputy Director Humphrey Dobson said: "Whatever your view of David Cameron's premiership it is saddening that he has chosen to highlight this deeply controversial issue as his legacy.
"We believe that if same-sex marriage had been put to a referendum it would have been defeated, but he chose not to give the British people a say on the redefinition of marriage."
Education Secretary Nicky Morgan, Home Secretary Theresa May and Lord Chancellor Michael Gove, all Christians, as well as former London Mayor Boris Johnson are all thought to be favourites to become the next Prime Minister.
