Archbishop Philip Tartaglia said Scottish bishops were "shamed and pained" by the suffering of those who had been harmed.
His apology followed the publication of a report from the McLellan Commission, which was set up to investigate abuse.
It called on the church to make an "unmistakeable and unequivocal" apology and "heal the hurt" of victims.
In response, Archbishop Tartaglia said: "As the president of the Bishops' Conference, and on behalf of all the bishops of Scotland, I want to offer a profound apology to all those who have been harmed and who have suffered in any way as a result of actions by anyone within the Catholic Church.
"Child abuse is a horrific crime. That this abuse should have been carried out within the church, and by priests and religious, takes that abuse to another level.
"Such actions are inexcusable and intolerable. The harm the perpetrators of abuse have caused is first and foremost to their victims, but it extends far beyond them, to their families and friends, as well as to the church and wider society."
He told survivors that the Catholic Bishops of Scotland were "shamed and pained by what you have suffered".
And he added: "We say sorry. We ask forgiveness.
"We apologise to those who have found the church's response slow, unsympathetic or uncaring and reach out to them as we take up the recommendations of the McLellan Commission."
The church asked Dr Andrew McLellan to lead a review of how it handles allegations of abuse following a series of scandals.
It took evidence from victims in a bid to improve support services and protect vulnerable children and adults.
Dr McLellan, a former moderator of the Church of Scotland, was tasked with coming up with proposals aimed at making the Roman Catholic Church in Scotland "a safe place for all".
The 11 review commissioners, who include a senior police officer, a journalist and an MP, were tasked with assessing the quality of support available to survivors.
It was not within the scope of the commission to investigate or adjudicate on current or historical allegations.
The commission made eight recommendations, including calling for support for survivors of abuse to be an "absolute priority".
It also said justice must be done for those who have been abused.
The report also recommended that the church's safeguarding policies and practices be completely rewritten and subject to external scrutiny.
It called for a consistent approach to dealing with allegations across Scotland and improved training for those in the church.
Mr McLellan said: "The Bishops' Conference of Scotland should make a public apology to all survivors of abuse within the church.
"An apology must be made in a way that is unmistakeable and unequivocal."
He added: "The Bishops have said from the outset that they will accept our recommendations.
"That means that three things will happen.
"First and most important a beginning will be made to heal the hurt and address the anger which so many survivors feel.
"Second, the Catholic Church in Scotland will begin to confront a dark part of its past and find some healing for itself.
"Third, a significant step will be taken in restoring public credibility for the Catholic Church."
Archbishop Tartaglia confirmed that the bishops had agreed to accept the recommendations in full.
The review was announced following a series of scandals.
The Church faced allegations of abuse at the former Catholic boarding school at Fort Augustus Abbey in the Highlands.
The former leader of the Catholic Church in Scotland, Cardinal Keith O'Brien, stepped down in February 2013 after admitting sexual misconduct.
The church said it would make public annual audits of all allegations received by the church.
It has published details of allegations dating from 2006 to 2012 and then for 2013.
The church also said it would instigate a retrospective investigation of historic allegations, dating back to 1947 - with work continuing on that.
It said any allegations uncovered as part of this, which had not been acted upon, would be passed to police for them to investigate.
The Roman Catholic Church in Scotland has issued a "profound apology" to victims of child abuse.