Cardinal Sean Brady has rejected calls to resign over claims that he helped cover up abuse by a paedophile priest.
On Tuesday, he said he would continue as leader of Ireland's Catholics.
It was revealed in March that the cardinal was present when children signed vows of silence over allegations against Father Brendan Smyth in 1975.
Survivors of Child Abuse spokesman John Kelly said the decision not to resign was an act by "a decent man who has not done the decent thing".
Abuse survivor Marie Collins said: "I think the church needs new leadership and I'm disappointed that we're not going to get it.
"It's not a question of revenge, it's a question of taking responsibility.
"We can't have change in the church here until everyone in the top realises how actions of the past decimated so many lives."
Maeve Lewis, of support group One in Four, also said Cardinal Brady should have resigned.
"If Cardinal Brady is genuine in his desire to oversee change in the Catholic Church, let him begin by challenging the adversarial, legalistic response that so many survivors continue to meet as they attempt to bring their experiences to the attention of diocesan and congregational authorities," she said.
Augustinian priest Father Iggy O'Donovan said Dr Brady was probably "better placed than many" to lead the Catholic church in Ireland.
"I'm looking at the larger picture, particularly the reaction from Rome in recent times where people who are holding far more senior positions than Cardinal Brady.
"I would see him as much better placed to do the job, so to speak, than some others."
On Tuesday, Cardinal Brady said he had decided to stay in his job "because I want to maintain the momentum towards better child safeguarding".
The Vatican must have decided that it would be better if the cardinal stayed as, what he's described as, the wounded leader, to try to restore trust and bring reconciliation inside the church.
I think it's significant that this new bishop he has asked to be appointed isn't being being deliberately introduced as a successor.
This will be an important appointment. We should look very carefully at this new archbishop or assistant bishop and expect that person to play a pretty big role in the future.
By Robert Piggot BBC religious affairs correspondent
He added: "Not only that, but also the momentum towards renewal of the faith which is essential, which is a big challenge here.
"I feel that I have to do what I'm called to do, what I'm placed here to do.
"I am also asking for help to do it, I'm not saying that I'm the greatest or anything like that."
Cardinal Brady has asked for a Vatican inspection of child protection procedures to include Armagh diocese and for an additional bishop in the diocese.
His comments follow the publication of the Annual Report of the National Board for Safeguarding Children in the Catholic Church in Ireland.
It found that nearly 200 new allegations of historical abuse have been reported to the Catholic Church's child protection watchdog since April 2009.
He said he would be appointing a director of child safeguarding for the diocese who would handle all future allegations of child abuse and report to civil authorities in both jurisdictions.
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