Faith schools should stop selecting pupils according to their religion and do more to serve the most disadvantaged children, the Runnymede Trust said today.
The charity, set up to promote good race relations, argues that faith schools could do much more to promote social cohesion.
Its report into the way faith schools operate in England, based on interviews over the past two years with more than 1,000 people, makes six key recommendations.
Faith schools should be "for the benefit of all in society rather than just some" and serve "the most disadvantaged in society".
"With state funding comes an obligation to be relevant and open to all citizens," the report concludes.
Children should have a much greater say in their education and religious education, covering all religions, should become part of the national curriculum.
Faith schools should value all young people, not just those of a particular religion, though religious identities should also be valued more highly in non-denominational schools, it suggests.
The report says faith schools should continue to play an important role in the education system as they offer diversity, a way to improve standards, and parental choice.
The proposals were launched today at a conference attended by the education minister, Sarah McCarthy-Fry. They are likely to reopen the debate over the role of faith schools in England's education system and whether they are too selective.
Earlier this year the schools secretary, Ed Balls, was accused of conducting a "witch-hunt" of faith schools after a government study found one in six schools broke admission laws - an accusation he strongly denied.
Rob Berkeley, the trust's deputy director and author of the report, said: "Given the importance of issues around cohesion, it's time for a shift, so that schools that are funded by taxpayers are responsive and reflect the needs of all pupils and not just those of a particular religion.
"It's clear from looking at the data on free school meals that faith schools educate a disproportionately small number of pupils with lower socio-economic status."
Faith schools should be able and willing to admit and recruit young people from a broad range of backgrounds, he said.
"People argue quite strongly that faith schools encourage indoctrination. We don't take that line, but we want to emphasise democracy in schools so young people have a say in how they are educated," he said.
Accord, a new coalition campaigning for inclusive schooling, which has concerns over how faith schools currently operate, particularly regarding selection, welcomed the report.
Rabbi Jonathan Romain, Accord's chairman, said: "The report comes from a respected neutral source so it will be much harder to dismiss.
"The debate has become so polarised that it's difficult to say anything against faith without being labelled a Richard Dawkins supporter."
He added: "Faith schools are signed up to serve society at large on paper but that's not quite the same are putting it into practice.
"We hope the government pays close attention to the report and turns its recommendations into legislation."
Barry Sheerman, chairman of the Commons' schools select committee, said: "We looked at admissions, and a large number of faith schools seemed to be enormously adept at keeping out poor and special education needs children.
"It shocked me that a whole group of schools were very good at not delivering education to the most needy."
He said the report would stimulate debate on the role of faith schools and how they should operate. "It's about time we had this serious debate."
According to the report, there are 6,900 faith schools in the state sector, the bulk of which (91%) are primaries. They make up a third of all maintained schools and most are Church of England and Catholic schools.
They sit alongside 11,106 non-denominational state primary schools and 2,756 secondaries and the government has said it does not want to expand the faith sector.
State faith schools in numbers
4,657 Church of England
2,053 Roman Catholic
82 other Christian
36 Jewish
8 Muslim
2 Sikh
1 Hindu