The Catholic primate of all-Ireland has said that he will not resign as Church leader despite revelations in the BBC's This World programme.
It found Cardinal Sean Brady had names and addresses of those being abused by paedophile priest Brendan Smyth.
However, he did not pass on those details to police or parents.
Cardinal Brady said he accepted he was part of "an unhelpful culture of deference and silence in society, and the Church".
"With others, I feel betrayed that those who had the authority in the Church to stop Brendan Smyth failed to act on the evidence I gave them," he said in a statement on Wednesday.
"However, I also accept that I was part of an unhelpful culture of deference and silence in society, and the Church, which thankfully is now a thing of the past."
The cardinal said he was "shocked, appalled and outraged" by Smyth and said he had trusted that those with the authority to act in relation to Smyth would treat the evidence seriously and respond appropriately.
He accused the BBC of exaggerating his authority in the programme.
"The commentary in the programme and much of the coverage of my role in this inquiry gives the impression that I was the only person who knew of the allegations against Brendan Smyth at that time and that because of the office I hold in the Church today I somehow had the power to stop Brendan Smyth in 1975.
"I had absolutely no authority over Brendan Smyth. Even my Bishop had limited authority over him. The only people who had authority within the Church to stop Brendan Smyth from having contact with children were his Abbot in the Monastery in Kilnacrott and his Religious Superiors in the Norbertine Order."
He added that he had worked with others in the Church to put these new procedures in place and looked forward to continuing that vital work in the years ahead.
Senior Vatican Prosecutor Monsignor Charles Scicluna has defended Cardinal Brady.
"My first point is that Fr Brady was a note taker in 1975, he did what he should have done. He forwarded all the information to the people that had the power to act," he said.
"My second point is that in the interest of the Church in Ireland, they need to have Cardinal Brady as the archbishop of Armagh because he has shown determination in promoting child protection policies. You need to have leaders who have learned the hard way and are determined to protect children."
The BBC investigation centres on a secret church inquiry in 1975 when a 14-year-old boy was questioned about abuse.
Smyth abused him and others in guesthouses on trips across Ireland.
In 1975, Cardinal Brady was a priest and teacher in County Cavan in the Republic of Ireland, when he was sent by his bishop to investigate a claim of child sexual abuse by a fellow priest.
That priest was later exposed as Ireland's most prolific paedophile, Father Brendan Smyth, who died in prison in 1997, one month into a 12 year prison sentence.
The first child to tell his parents about the abuse was 14-year-old Brendan Boland.
The man tasked with the secret church investigation that followed would later become the most senior priest in Ireland.
Sean Brady's role in the affair became clear in 2010, when it became known that he had been present when the abused boy was questioned.
He claimed, however, that the boy's father had accompanied him, and described his own role as that of a note-taker.
However, the BBC This World investigation has uncovered the notes Cardinal Brady took while the boy was questioned.
The child's father was not allowed in the room, and the child was immediately sworn to secrecy.
What Cardinal Brady failed to tell anyone in 2010 was that Brendan Boland had also given him and his colleagues the precise details of a group of children, some of whom, were being abused by Smyth.
Cardinal Brady did interview one of them and swore him to secrecy.
This World spoke to all of the children who Brendan Boland had identified; they all told the programme that to the best of their knowledge none of their parents or families were warned in any way about the paedophile Brendan Smyth.
Four of them had been abused by Smyth. Two of them continued to be abused after the 1975 inquiry.
One of them - originally from Belfast - told the programme that Smyth continued to abuse him for another year.
He also said Smyth abused his sister for a further seven years and then in turn, his four younger cousins, up to 1988.
Cardinal Brady did consider his position as Primate of all-Ireland when his role in the secret inquiry was first exposed.
The Catholic Church has said that "the sole purpose of the oath" signed by Brendan Boland in Cardinal Brady's presence was "to give greater force and integrity to the evidence given by Mr Boland against any counter claim by Fr Brendan Smyth".
The church also points out that in 1975, "no state or church guidelines for responding to allegations of child abuse existed in Ireland".
On Wednesday night, Brendan Boland told BBC's The Nolan Show Cardinal Brady should resign.
"I can't understand why he can't see his failings," he said.
"Resign. Own up to your failings."
Irish Prime Minister Enda Kenny said it was a matter for Cardinal Brady to reflect on his decision following the BBC documentary.
Irish Justice Minister Alan Shatter said the programme's revelations were "tragic and disturbing".
Abuse victims' campaigner Marie Collins, who was raped at the age of 13 by a hospital chaplain in Dublin, said Cardinal Brady should resign.
"I'm amazed no bishops have come out and said he should go," she said.
"We have priests and theologians being silenced by the Vatican - they can act against people whose views they feel are liberal, but they will not act against someone who not only endangered children but let them be abused.
"If Cardinal Brady came out and espoused the view that women should be ordained, he'd be gone within hours."
Andrew Madden, abuse survivor and author of a memoir, said: "He's not a wounded healer - he's a spineless self-serving careerist and that's why he kept his mouth shut all the years Brendan Smyth was abusing children"
Gary O'Sullivan, editor of the Irish Catholic newspaper, said Cardinal Brady had questions to answer.
"If a child can see the need to save other children, how come priests, ministers of Christianity, cannot have the same awareness?" he said.
"If he wants to stay in this leadership position, he should show leadership and come out and answer these questions because this culture of silence failed children."
This World: The Shame of the Catholic Church was first broadcast on BBC One Northern Ireland on Tuesday. The programme is being shown on BBC Two at 21:00 BST on Wednesday.