The Voice of the Faithful has also challenged the pope to order an Ireland-wide inquiry covering every diocese to examine further cases of priests abusing children.
In the letter, the group says "accountability cannot be achieved while so many bishops and archbishops, who have knowingly over a considerable period of time permitted this tragedy to persist, continue in office".
The group, which also has branches in North America, Australasia and Europe, asks the pope to order an island-wide inquiry into each diocese. So far the church in Ireland has resisted demands for an investigation covering all 26 Catholic dioceses.
Last week's report, carried out by Irish judge Yvonne Murphy, identified four former archbishops of Dublin as failing to report their knowledge about child sex abuse to the Garda.
The Vatican and the papal nuncio in Dublin were both singled out for criticism in the Murphy report, accused of ignoring requests from the judge and her commission for information about abuse in the Dublin diocese.
The Voice of the Faithful letter tells the pope that "repeating the tragedy of Boston, Philadelphia, Los Angeles, Ferns, Cloyne, Sydney and so many other dioceses worldwide, four archbishops of Ireland's most populous diocese have behaved in a manner that facilitated the deepest psychological, emotional and spiritual trauma to many children.
"They have also endangered gravely the divine mission of the church, as well as the souls once more repelled from it. The time to act is now. The secrecy must come to an end. We are convinced that this programme cannot be speedily achieved in Ireland or elsewhere without the deployment of the full authority of your own office."
Sean O'Conaill, Voice of the Faithful's acting co-ordinator in Ireland, said that if the pope ignored their calls, "we will be forced to a conclusion that will be fatal to Catholicism globally: that the papacy also puts the Catholic clerical institution before the interests of children.
"To deny that change is now necessary in the way the church governs itself is to condemn other children of the church to the same trauma and to condemn the rest of the church to endless derision and scandal.
"Catholicism cannot survive this. To argue that God supports the present church system is to argue God approves of child abuse – and that is blasphemy."
Voice of the Faithful grew out of the abuse scandal that rocked the Catholic church in the United States and led to the sacking of Cardinal Law, the head of the church in Massachusetts.
Meanwhile one of the main victims' organisations, Irish Survivors of Child Abuse, called yesterday for any national inquiry to include the alleged role of secret Catholic societies in covering up the scandal.
"We know that certain politicians were connected to 'Opus Dei' and that organisation's reach is very very long. It extends into many fields of public life in Ireland. The question any inquiry on a national level should ask is, did these societies or their members exercise any influence on the decisions not to pursue the abusers in the interest of the church's reputation?"
Irish SOCA co-founder Patrick Walsh said that despite a 10-year battle to expose the truth about abuse both in dioceses and church-run institutions, he was still shaken by the content of the Murphy report: "We wondered why at the time the Irish hierarchy were so hostile towards us. Now we know they were protecting themselves and their religious orders. We see the fuller picture from these three reports, the upshot being that we have a church disgraced, totally and completely."
However, Walsh praised the role of the current Dublin Archbishop Diarmuid Martin in helping the victims and facilitating the Murphy Commission: "He has clean hands. Unlike other bishops and cardinals, he is a man of honour who opened the books for the commission, and for that he deserves everyone's praise."
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