"All endeavors at evangelization and other great works will be dependent upon our ability to own our crimes and failings and to make the protection of children and vulnerable adults our No. 1 priority", O'Malley said in a statement read out to a safeguarding panel at the World Meeting of Families.
But Francis offered no such structures or sanctions in a letter he penned on the eve of his Irish visit to the world's 1.2 billion Catholics, vowing only to spare no effort to fight the abuse problem.
The pope later held an hour and a half meeting with victims of abuse at the hands of clergy or in Church-run institutions, including Marie Collins, who was, at the age of 13, abused by a priest while being treated in a hospital in Dublin.
Pope Francis marked the first papal visit to Ireland in 39 years by acknowledging that the failure of Church authorities to adequately address "repugnant" clerical child abuse crimes there remains a source of shame for the Catholic community. "I myself share these sentiments".
In a strongly-worded speech at the start of the pope's visit, Varadkar said the abuses in Ireland were "stains" on the Catholic Church, the state and Irish society as a whole.
Varadkar, an openly gay leader and a symbol of Ireland's liberalising culture, demanded from Pope Francis "that from words flow actions" for victims in a strongly-worded speech.
He added: "In place of Christian charity, forgiveness and compassion, far too often there was judgment, severity and cruelty, in particular towards women and children and those on the margins".
"Holy Father, I ask that you use your office and influence to ensure this is done here in Ireland and across the world", Varadkar said.
Paul Redmond, who was born in a Catholic mother and baby home, said the pope had described the cover-up of abuse as "caca", or excrement.
On Tuesday the grand jury in Pennsylvania issued a report that 300 Catholic priests across the U.S. state sexually abused children over seven decades, protected by church leaders. He could have talked to us all in a way that was blunt, that was clear, that was frank, that was human and that was accessible.
Hundreds of thousands are expected to welcome the pope, but there will also be demonstrations and vigils by survivors of clergy sex abuse in a country where attitudes on the Catholic Church are shifting.
They set children's shoes on the pavement, each representing a child who had been abused by priests.
Later in the day he is due to meet newlyweds and couples about to marry at St Mary's Pro cathedral in Dublin.
The Coalition said that Pope Francis agreed to include a message that natural mothers who lost their children to adoption did nothing wrong in his Phoenix Park mass on Sunday.
He will begin the two-day visit by meeting Ireland's President Michael D. Higgins and Varadkar, who has promised to challenge the pope to do more in dealing with the abuse crisis.
More than three-quarters of the Irish population flocked to see Pope John Paul II in 1979 at a time when divorce and contraception were illegal.
The country voted for constitutional change on same sex-marriage in 2015 and voted overwhelmingly to overturn its strict abortion law in May. Organizers of the "Say Nope to the Pope" protest said on Facebook they hoped to show solidarity to abuse victims and "show the Church they don't have the control they used to".
The Vatican was rocked this month by a devastating USA report into child sex abuse that accused more than 300 priests in the state of Pennsylvania of abusing more than 1,000 children since the 1950s.