Helen McGonigle, a US lawyer who was abused by Fr Brendan Smyth in the 1960s, has called on Cardinal Seán Brady to resign.
Ms McGonigle called for a public criminal investigation into the matter and for the Cardinal to be prosecuted if there is evidence that he failed to report crimes to the police.
The BBC's ''The Shame of the Catholic Church'' documentary claimed that Cardinal Brady did not pass on information about abuse to families of children involved.
Speaking on RTÉ's News at One, Ms McGonigle accused Cardinal Brady of only thinking in terms of canon law.
She said: "He didn't fulfil the function of a human being, a man in the secular world or follow secular law. And he can't see that.
"It's as if the man has been completely brainwashed and he can't think outside of his role as a cleric. That's the fundamental problem.
"So when he says "I did act". Oh, I'd agree he acted. He acted in the most horrible of ways, in the ways that the Catholic Church required.
"I want him to resign. I plead for both the Republic and Northern Ireland to conduct a public criminal investigation into this whole matter. And that it be made public."
She said US officials were beginning to charge clerics for failing to report allegations to police and said the same needed to happen in Ireland.
"[Cardinal] Brady needs to be investigated and if the evidence is there, he needs to be prosecuted and charged."
Elsewhere, the Catholic Communications Office rejected reports that Cardinal Brady was willing to resign two years ago over the affair.
A report in today’s Irish Independent says that the Vatican rejected an offer from Cardinal Brady to step down.
Brady controversy about 'acceptance of responsibility'
Taoiseach Enda Kenny said it is not for him to determine who leads the Catholic Church, or any other organisation in Ireland.
Mr Kenny said: "Clearly the controversy surrounding Cardinal Brady is about the acceptance of responsibility and it is the responsibility of the State to enact laws to protect children."
He said the Government had already appointed a senior Minister for Children and published legislation on mandatory reporting when it comes to the sexual abuse of children.
Tánaiste Eamon Gilmore denied that some politicians are trying to hound Cardinal Brady out of office.
He said it was the responsibility of the Church and not the Government to decide who remains in or leaves a position.
But he said it is the Government's duty to ensure there is adequate protection for children.
Meanwhile, one of the country's leading theologians has said Cardinal Brady has lost his moral credibility and should resign as the Catholic Primate of All Ireland.
Fr Vincent Twomey, a retired professor of moral theology at St Patrick's College in Maynooth, said there were issues arising from the current controversy that the Catholic Church must address internally.
The BBC documentary alleged that claims made by a boy in 1975 - to a church inquiry - were not passed on to parents of other victims or to gardaí or police.
Fr Twomey said that he thought Cardinal Brady should now step down for the good of the church.
Minister for Social Protection Joan Burton has also called for the Cardinal to resign.
Speaking on RTÉ's Today with Pat Kenny, she said: “He was a man in his middle 30s at the time who was a Doctor of Divinity.
"You could say at that stage that people like that were tremendously naive. But he was highly educated. He also held a position in a school.
"I personally think that he needs to reflect on his position, and were he to ask me for my view on a personal basis, I would say his position is not really sustainable.”