Helen McGonigle, who was molested by the Rev. Brendan Smyth in Rhode Island in the 1960s, said she was "outraged" by Brady's response to "The Shame of the Catholic Church," a BBC documentary broadcast this week.
The documentary alleged that Brady failed to report the sexual abuse of children he learned of in the 1970s. The cardinal said the report exaggerated his role and that he didn't help hide sexual abuse by priests. He has refused calls for his resignation.
McGonigle said Brady had shown "arrogance and insensitivity." She said his main "duty in life" was to protect children and his failure to act was "unforgivable."
McGonigle called for Brady's resignation and for an investigation into possible criminal charges.
Smyth was arrested in 1994 on other charges of child abuse and sentenced to 12 years in prison after admitting to molesting dozens of children. Authorities said his actions were widely covered up by other members of the Catholic Church.
Minister for Social Protection Joan Burton and the Rev. Vincent Twomey, a leading theologian, also have called for Brady's resignation
"He was a man in his middle 30s at the time who was a doctor of divinity," Burton said. "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."
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