The pope acknowledged he had "hurt" people by suggesting there was no "proof" against Juan Barros, who is accused of covering up another priest's abuse of boys.
"The day they bring me proof against Bishop Barros, then I will speak. There is not a single piece of proof against him. Everything is slander. Is this clear?" the Argentine pontiff said abruptly on Thursday when questioned by journalists in Chile.
The pope also shocked many by giving Barros a hug during his visit.
"I have to present my apologies because the word 'proof' has hurt so many victims of abuse," Francis said at a press conference on the flight back to Rome.
"To hear the pope tell them to their faces, 'bring me a letter with the proof', it's a slap and I realise now that my expression was unfortunate," he said.
However, Francis said the Vatican had investigated Barros, without finding "any element to condemn him".
Running the risk of displeasing people in Chile, where Catholicism is falling, he said he was "convinced" of the bishop's innocence.
"You tell me that there are victims, but I did not see them," he said.
Zero-tolerance
On the plane, he said he should have used the term "incriminating evidence".
"The word 'proof' was not the best to bring me closer to an aching heart," he conceded. "I know there are many people who are abused and cannot prove anything."
"I know how much they suffer," insisted Francis, who met privately in Chile with two victims of paedophile priests. According to the Vatican, he "prayed and cried" with them.
"The tragedy of the victims of abuse is horrible, horrible. Two months ago, I was in contact with a woman who was a victim 40 years ago.
"Now married with three children, the woman hadn't spoken about it since, because in the hand of the priest, she saw the hand of her sexual abuser," the pope said.
On Saturday, Cardinal Sean Patrick O'Malley -- who heads a paedophile commission undergoing an overhaul in the Vatican -- emphasised Francis's sincerity when he advocated a zero-tolerance policy against paedophilia in the Catholic Church.
But in an unusual criticism of the pope, he had judged as "understandable" that his words could have caused "great pain".
"His statement was very fair," the pope said on the plane.
"He recalled everything I did and what I do, and what the Church does, and then he spoke of the pain of the victims."
The pope's six-day visit to Chile and Peru end by him celebrating an open-air Mass on Sunday for more than a million people in the Peruvian capital, Lima.
In Peru, he also spoke out against political corruption in the region and dangers to indigenous people.
In Chile, he visited a women's prison and prayed with survivors of Augusto Pinochet's dictatorship.
