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Tory backbencher Peter Bone said Mr Clegg had insulted "millions of people with deep convictions of religion and conscience".
He insisted the Lib Dem leader had to "apologise profusely" or resign.
"I don't see how that could have got published without it being the view of the Deputy Prime Minister," he said.
"He has got to rapidly get out there on the airwaves apologising," Mr Bone said.
"It is clear what he thinks. There is no way that the Deputy Prime Minister of our country can be associated with that language."
Former Archbishop of Canterbury Lord Carey has said the mooted remarks by Nick Clegg were "very offensive".
"If he persists in taking that view I and others would be very offended," he told the BBC.
"To be called a bigot is a very offensive statement and I would ask him to recall it... because there are issues here that demand very serious debate."
The coalition's pledge to introduce same-sex civil marriage by 2015 has been criticised by religious groups and provoked deep unease in Conservative ranks.
Deputy Prime Minister Nick Clegg said there had been 228,000 responses to the consultation since it was launched in June, and the government would set out its next move by the end of the year.
David Cameron has signalled that MPs will be given a free vote on the issue, but stressed his personal commitment.
The Deputy Prime Minister Nick Clegg responded tonight after withdrawing comments where opponents of equal marriage were branded as "bigots".
Deputy Prime Minister Nick Clegg was embarrassed today after being forced to withdraw comments in which he branded opponents of equal marriage as "bigots". The first extracts released early by his office said:
But it was quickly re-written to: