Cardinal Keith O'Brien, the former leader of the Catholic Church in Scotland, has admitted his sexual conduct has at times "fallen beneath the standards expected of me".
He apologised and asked forgiveness from those he had "offended".
In a statement, he also apologised to the Church and the people of Scotland.
The cardinal resigned last Monday after three priests and a former priest had made allegations of improper behaviour against him dating back to the 80s.
Cardinal O'Brien was Britain's most senior Roman Catholic cleric until he stood down as the Archbishop of St Andrews and Edinburgh.
Life in retirement
In announcing his resignation, he also said he would not take part in the election for a successor to Pope Benedict XVI, known as the conclave.
Sunday's further statement, issued through the Roman Catholic Church in Scotland, read: "In recent days, certain allegations which have been made against me have become public. Initially, their anonymous and non-specific nature led me to contest them.
"However, 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.
"To those I have offended, I apologise and ask forgiveness. To the Catholic Church and people of Scotland, I also apologise.
"I will now spend the rest of my life in retirement. I will play no further part in the public life of the Catholic Church in Scotland."
The cardinal - who had initially said he was taking legal advice when the allegations against him were made public - had been due to retire later this month when he turned 75.
The former priest and three current priests from the diocese of St Andrews and Edinburgh complained to the Pope's representative to Britain, Archbishop Antonio Mennini, in early February about what they alleged had been inappropriate behaviour towards them in the 1980s.
Earlier on Sunday, the ex-priest told the Observer he had been disappointed by the "cold disapproval" he had faced from the Church for "daring to break ranks".
He also claimed to have gone public only because he feared the matter was in danger of being swept under the carpet by the church.
The BBC's correspondent in Rome, David Willey, said the news would no doubt affect the mood at the papal conclave at the Vatican.
He added: "The Rome gathering is already overshadowed by allegations of scandal, intrigue and betrayal among the Vatican hierarchy.
"The credibility of the Roman Catholic Church will be further damaged by Cardinal O'Brien's confession of inappropriate sexual behaviour."
Last week, the Observer reported that the former priest had claimed Cardinal O'Brien had made an inappropriate approach to him in 1980, after night prayers, when he was a seminarian at St Andrew's College, Drygrange.
A second statement from another complainant said he had been living in a parish when he was visited by Cardinal O'Brien, and inappropriate contact had taken place between them.
A third complainant alleged he had faced what he described as "unwanted behaviour" by the cardinal in the 1980s after some late-night drinking.
The fourth complainant claimed the cardinal had used night prayers as an excuse for inappropriate contact, the newspaper said.
Cardinal O'Brien, who was born in Ballycastle, Co Antrim, had been the Archbishop of St Andrews and Edinburgh since 1985.
He has been an outspoken opponent of plans to legalise same-sex marriage in Scotland and was named "bigot of the year" by gay rights charity Stonewall last year.
In an interview with BBC Scotland shortly before the allegations against him were made public, the cardinal said he believed priests should be able to marry and have children if they wished.
He said it was clear many priests struggled to cope with celibacy.