Dr Noel Treanor made the comments to a congregation in Carryduff
The Bishop of Down and Connor has apologised to all victims of child sex abuse by the Catholic church.
Dr Noel Treanor was speaking after a report last week said thousands had been abused over 60 years.
He said the report was "heartbreaking" and had recorded cruelty and abuses which were "criminal and sinful."
But Dr Treanor said the church was addressing the "evil", adding that the Down and Connor diocese now had robust child protection measures in place.
"I state my sorrow, shame and visceral pain in the face of these and all abuses inflicted on children and vulnerable adults, whenever they took place, wherever they are perpetrated," he told a congregation in Carryduff on Sunday.
"I apologise on behalf of the church to all who are victims of abuse on the part of those who professed to care for them, or minister to them, in the name of Christ.
"I apologise, too, for the failure of those in positions of leadership in the Church to deal with the abusers."
The victims of child abuse by religious orders were among 35,000 children who were placed in a network of reformatories, industrial schools and workhouses until the early 1990s.
Abuse at Catholic institutions was investigated
More than 2,000 people told the Commission to Inquire Into Child Abuse they suffered physical and sexual abuse as children in the institutions.
The commission found that sexual abuse was "endemic" in boys' institutions, and church leaders knew what was going on.
Dr Treanor said: "As we grasp the extent and dimensions of this evil that has been at work within the church, we have to recognise that as a church in particular, and as society, and as individuals, we stand in need of chastening our moral and personal radar.
"Anger, indifference, denial, washing one's hands of guilt, like Pilate, if partly comprehensible as reactions, will not suffice on the part of anyone."
The Irish deputy prime minister called the abuse of children in Catholic-run institutions as one of the "darkest chapters" in Irish history.
The leader of the Catholic Church in Ireland, Cardinal Sean Brady, said anyone responsible for the abuse should be held to account.
The report, nine years in the making and covering a period of six decades, also found government inspectors failed to stop beatings, rapes and humiliation.
The findings will not be used for criminal prosecutions.