O'Brien stepped down from his post as Archbishop of St Andrews and Edinburgh last week after three priests and a former priest complained to the Vatican about inappropriate behaviour towards them going back 30 years.
O'Brien had contested the claims.
On Saturday he claimed in the Observer he was "disappointed" by the Church's response and that "if they could crush me, they would".
In a statement issued by the Catholic Church in Scotland on Sunday night however, he apologised and asked forgiveness from those he had "offended".
"I wish to take this opportunity to admit that there have been times that my sexual conduct has fallen below the standards expected of me as a priest, archbishop and cardinal," the statement said.
Cardinal O'Brien said he will spend the rest of his life in retirement and will play no further part in the public life of the Catholic Church in Scotland.
It is understood he is currently out of the country.
The cleric, who came under the spotlight earlier this year after being named 'bigot of the year' by gay rights charity Stonewall, had already said he would not be joining the conclave to choose a successor to Pope Benedict XVI.
The Cardinal was the only British Roman Catholic cleric able to vote in the upcoming conclave to elect a successor to Pope Benedict XVI following his earlier decision to resign.
O'Brien said: "I do not wish media attention in Rome to be focussed on me - but rather on Pope Benedict XVI and on his successor."
The cardinal, who initially said he was taking legal advice when the allegations against him emerged, had been due to retire later this month when he turned 75.
One of the alleged victims criticised the Church's response to the scandal.
They told the Observer: "There have been two sensations for me this week. One is feeling the hot breath of the media on the back of my neck and the other is sensing the cold disapproval of the church hierarchy for daring to break ranks. I feel like if they could crush me, they would.
"The vacuum the church has created has allowed whimsy and speculation to distort the truth, and the only support I have been offered is a cursory email with a couple of telephone numbers of counsellors hundreds of miles away from me."
Another priest called for the Catholic Church in Scotland to be more open in dealing with complaints.
Father John Robinson told the Sunday Mail: "If the Catholic Church in Scotland is to heal itself, we need transparency and understanding.
"We need to learn lessons from the mistakes we have made in the past and move on to become a more loving and understanding church which does not condemn victims or even abusers."
The allegations against the cardinal emerged just days after he called for the Catholic Church to end its celibacy rule for the priesthood.
He said that many priests struggle to cope with celibacy and should be allowed to marry if they wish.
The cardinal, who was born in Ballycastle, Co Antrim, had been the Archbishop of St Andrews and Edinburgh since 1985.
Ordained as a priest in 1965, he was proclaimed a cardinal by Pope John Paul II in October 2003.
His absence from the conclave to elect the next pope means that the church in Britain has no vote in the process.
The archbishop who is temporarily replacing Cardinal O'Brien recently spoke of the "pain and dismay" he shares with worshippers as he took Mass for the first time since being appointed.
Archbishop Philip Tartaglia told those gathered at St Mary's Cathedral in Edinburgh last week that he understands that the archdiocese is in a "state of shock for the loss of its shepherd" following the cardinal's resignation.
The pope appointed the Archbishop of Glasgow to govern the Roman Catholic Archdiocese of St Andrews and Edinburgh until a permanent replacement is chosen.