Dr Rowan Williams made the comments in an interview with Andrew Marr to be broadcast on BBC Radio 4's Start the Week programme.
In an uncharacteristic breach of ecumenical protocol, the Archbishop said in the interview: "I was speaking to an Irish friend recently who was saying that it's quite difficult in some parts of Ireland to go down the street wearing a clerical collar now.
"And an institution so deeply bound into the life of a society, suddenly becoming, suddenly losing all credibility - that's not just a problem for the Church, it is a problem for everybody in Ireland."
Dr Williams later called the Catholic Archbishop of Dublin Diarmuid Martin to express his "deep sorrow and regret for any difficulties which may have been created" by his comments and to insist that "nothing could have been farther from his intention than to offend or criticise the Irish Church".
Archbishop Martin said he was "stunned" to hear Dr Williams' remarks and that he had "rarely felt personally so discouraged" as he did when he heard them.
"I still shudder when I think of the harm that was caused to abused children. I recognise that their Church failed them," he said in a statement.
"But I also journey with those – especially parents and priests - who work day by day to renew the Catholic Church in this diocese and who are committed to staying with their Church and passing on the faith in wearying times.
"Archbishop Williams' comments will be for them immensely disheartening and will challenge their faith even further."
He added that those working for renewal in the Church did not need such a comment on Easter weekend and "do not deserve it".
The controversy comes after Jewish groups and the media accused a senior Catholic figure of comparing criticism of the Pope in recent weeks to anti-Semitism suffered by Jews.
In a Passion of the Lord service in St Peter's Basilica on Friday, preacher to the Pope, Father Raniero Cantalamessa quoted a letter from a Jewish friend who had written to him condemning the "attacks" on the Pope over his handling of a child abuse case in the US in the 1990s.
He quoted: "The use of stereotypes and the passing from personal responsibility and guilt to a collective guilt remind me of the more shameful aspects of anti-Semitism."
The Vatican later issued a statement stressing that it did not regard the criticism of the Pope "to be in any ways similar to anti-Semitism" and that Fr Cantalamessa had "intended only to share an expression of solidarity from a Jewish brother".