The Deputy Prime Minister moved to limit damage after the jibe appeared in an early draft of a speech mistakenly released to the media.
In the letters to the Archbishops of Canterbury and Westminster, Mr Clegg said: “Those extracts were neither written or approved by me.
“They do not represent my views, which is why they were subsequently withdrawn.
“While I am a committed advocate of equal marriage, I would never refer to people who oppose it in this way. Indeed, I know people myself who do not support equal marriage and, although I disagree with them, clearly I do not think they are bigots.
“Nor do I think it is acceptable they, or any one else, are insulted in this way. My views on this issue are no secret, but I respect the fact that some people feel differently to me about marriage, often because of their religious beliefs.
“I hope this explanation helps clarify what happened yesterday as well as my position, and I hope that the serious error that occurred will not cause lasting offence.”
The extracts were issued to journalists in advance of a star-studded reception held in central London to celebrate a consultation on the policy.
The text complained that “continued trouble in the economy gives the bigots a stick to beat us with, as they demand we ‘postpone’ the equalities agenda”.
However, shortly afterwards a corrected version was issued replacing “bigots” with “some people”.
As aides tried to quell an angry backlash, the Liberal Democrat leader told guests at the event he would never use such “insulting” language.
In the letters to Archbishop of Canterbury Dr Rowan Williams and Archbishop of Westminster Vincent Nichols today, Mr Clegg said he wanted to “clear up any confusion”.
But he stopped short of apologising for the “serious error” by his office.
The gaffe echoed an incident during the 2010 election campaign when then prime minister Gordon Brown labelled Rochdale pensioner Gillian Duffy a “bigoted woman”, a comment that was inadvertently picked up by a microphone.
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 - with some ministers suggesting they may not support it in Parliament.
David Cameron has signalled that MPs will be given a free vote, but stressed his personal commitment to the change.