A senior Archbishop has urged the Catholic Church in Ireland to move out of "denial" and take a "reality check" after the vote in favour of same-sex marriage.

The Catholic Church opposed the change and at one point four bishops spoke out in unison calling on all Catholics to vote "no".

The overwhelming majority that voted in favour, in spite of the Church's call to hold fast to traditional values that marriages can be only between a man and a woman, is being regarded as a sign of the Church's waning influence in the wake of the revelations about child abuse in religious institutions.

The results became known on a day marked by traditional Irish early summer weather of sunlight breaking through rain, drizzle and clouds. Pictures of rainbows that appeared nationwide as a result were posted on Twitter, Facebook and other social media by hundreds of people celebrating the vote.

With the characteristic humility for which he has become known, the widely-respected Archbishop of Dublin, Most Rev Diarmuid Martin, accepted defeat gracefully and said the vote illustrated the need to reconnect with young people if there was to be any hope of restoring the Church's moral authority.

He told RTE: "We have to stop and have a reality check, not move into denial of the realities. We won't begin again with a sense of renewal, with a sense of denial.

"I appreciate how gay and lesbian men and women feel on this day. That they feel this is something that is enriching the way they live. I think it is a social revolution."

He voted "no", in line with Catholic teaching. While believing gay rights should be respected, he believes this should be done "without changing the definition of marriage".

He said: "I ask myself, most of these young people who voted yes are products of our Catholic school system for 12 years. I'm saying there's a big challenge there to see how we get across the message of the Church."

The failure of Catholic education to carry the message against gay marriage despite the Church's teaching shows the fundamental problem facing the Catholic Church. The same issue also faces Protestant and other churches in western nations, including the Church of England, which has been "protected" against having to engage with gay marriage in the recent UK law.

Young people educated in religious and well as secular schools are being taught to respect equality, the right to life and all that life offers including sexual intimacy and the dignity of every human being, whatever their sex or sexual orientation. It is increasingly difficult, if not impossible, for young people to square this secular gospel believed by the majority with the religious doctrine practised by an ageing minority.

There is growing recognition among church leaders that the Church's stand against gay marriage in such a post-enlightenment educational culture is failing to hold back the tide.

The prime minister, David Cameron, was among the many committed Christians who welcomed the vote. He tweeted: "Congratulations to the people of Ireland, after voting for same-sex marriage, making clear you are equal if you are straight or gay."

Nearly 1.95 million people voted, a turnout of 61 per cent, higher than the 56 per cent who voted to ratify the 1998 Good Friday agreement. The first same sex marriages are expected to take place later this year.