The critics, who penned an open letter, included more than 50 writers, scientists, broadcasters and academics, who accused Mr Cameron of risking sowing "alienation and division" in a country where most people did not describe themselves as Christian.
The signatories included authors Philip Pullman and Sir Terry Pratchett, broadcasters Dan Snow and Nick Ross, philosopher AC Grayling and human rights activist Peter Tatchell.
Professor Jim Al-Khalili, a theoretical physicist and science broadcaster who organised the letter, said the PM's comments were part of a "disturbing trend".
"Politicians have been speaking of our country as 'a Christian country' with increasing frequency in the last few years; not only is this inaccurate, it's a wrong thing to do in a time when we need to be building a strong shared identity in an increasingly plural and non-religious society," said Mr Al-Khalili, who is president of the British Humanist Association.
But Mr Straw, who is a Christian himself, said he did not accept this analysis.
"There has to be a clear understanding that this is the UK and there are a set of values, some of which I would say to the letter writers are indeed Christian-based, whether they like it or not, which permeate our sense of citizenship," insisted the MP for Blackburn.
The criticism of Mr Cameron came after he wrote an Easter article for the Church Times, in which he expressed the desire to infuse politics with Christian ideals and values.
Downing Street came to the PM's defence, saying he had made clear how "this was not to say in any way that to have another faith -or no faith - was somehow wrong".
Religious groups also backed the PM. The Hindu Council UK said it was "very comfortable" with his description of Britain as a Christian country while the Muslim Council of Britain said it regarded the UK as a largely Christian nation.