More than three in five voters support David Cameron's wish to introduce gay marriage, according to a poll conducted for the Guardian. The strong backing for a change in the law comes after the archbishop of Westminster queried the democratic legitimacy of the coalition plans.

Vincent Nichols, head of the Roman Catholic church in England and Wales, used a strident Christmas Day message to blast the "shambolic" process that could soon put provision for same-sex weddings on the statute book. "There was no announcement in any party manifesto; there's been no green paper; there's been no statement in the Queen's speech. And yet here we are on the verge of primary legislation," Nichols told the BBC.

The plans also came under fire from a high court judge, who said the government should instead be looking at the "crisis of family breakdown". Sir Paul Coleridge said too much time and energy had been put into the debate on gay marriage for "0.1% of the population".

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The ICM poll conducted just before Christmas found 62% of voters now support the proposals, with half this number – 31% – opposed. Most previous polls have found opinion leaning the same way, although the two-to-one margin revealed on Wednesday is particularly emphatic. An ICM online survey for the Sunday Telegraph in March asked the identical question – which expressly reminds people that the option of civil partnerships already exists for gay couples – and established a 45%-36% lead for the reformers.

That significant hardening of opinion during the year will encourage Cameron, whose embrace of gay marriage has proved controversial, not only with religious leaders but also with the Tory backbench. And the new poll reveals a particularly significant swing towards the reform among the Tory base.

Although Labour and Liberal Democrat supporters remain more likely to support gay marriage, with respective majorities of 67% and 71%, there is now also a majority among Conservative supporters. Among those who voted Tory in 2010, gay marriage now enjoys 52%-42% backing, a big turnaround from ICM's survey in March, which recorded 50%-35% opposition from 2010 Conservative voters.

Both men and women support gay marriage, although the majority is bigger among female voters, 65% of whom support gay marriage, compared with 58% of men. Gay marriage is backed by 60%+ majorities across every nation and region, the 74% majority recorded in Wales being the most emphatic. There is a pro-gay-marriage majority, too, in every social class – although the majority is somewhat smaller in the DE class, which contains the lowest occupational grades. Fifty-one per cent of this group is in favour of the change, as opposed to 68% in the C1 clerical grade, which emerges as the most enthusiastic.

Sharper differences emerge when the results are analysed across the age ranges. The over-65s resist the proposal, by 58% to 37%, but support is progressively stronger in younger age groups. The pro-reform majority is 64% among 35-64s, 75% among 25-34s, and an overwhelming 77% among 18-24s.

The results will encourage the view of Cameron and the deputy prime minister, Nick Clegg, who has also set significant store by the proposal, that they are on the right side of history. But with opinion moving so fast in favour of gay marriage, enthusiasts on Wednesday demanded to know why the government had provided for such sweeping exemptions for religious groups. Among other things these provide for a wholesale exemption for the Church of England, such that – were the currently opposed C of E to embrace gay marriage at some later date – fresh legislation would be required to extend the reform to it.

As the Guardian poll findings were released, the Liberal Democrat minister Lynne Featherstone said that Roman Catholic leaders who were attacking the government's gay marriage plans were being "shameful".

In a Boxing Day post on her blog, she singled out comments from Nichols and from the Rt Rev Mark Davies, the Roman Catholic bishop of Shrewsbury, and said they wrong to suggest that churches would be forced to conduct gay weddings and wrong to dismiss the policy as illegitimate on the grounds that the coalition did not have a mandate for it.

"It is very disappointing that religious leaders who object so forcefully to equal marriage seem to have so little faith in their own beliefs," said Featherstone, an international development minister who used to be minister for equalities.

"If their religious beliefs are that marriage can only be between a man and a woman – they should have the confidence in their flocks to believe that too. And if it is their own flocks' potential for disagreeing with them that is their real fear – then that is a matter for religious leaders and their congregations to sort out."

• ICM Research interviewed a random sample of 1,002 adults aged 18+ by telephone on 19-23 December 2012. Interviews were conducted across the country and the results have been weighted to the profile of all adults. ICM is a member of the British Polling Council and abides by its rules.