The National Catholic Register reported that Viganò's letter was "simultaneously released" to the Register and other media.
The letter sent shock waves through the Catholic world as Francis, finishing a two-day trip in Ireland, was begging forgiveness for the "scandal and betrayal" caused by church-related abuses.
Francis and other church leaders are facing a bitterly polarised Catholic Church, and some Francis critics, including Viganò, are calling for the Pope to step down.
The Vatican had no immediate comment. McCarrick's lawyer, Barry Coburn, declined to comment.
The letter was the latest dramatic development stemming from a fresh wave of allegations related to clergy sex abuse and its cover-up. Rumours that had swirled for decades about McCarrick exploded in June when Pope Francis suspended the cardinal.
Last month, McCarrick, facing credible allegations of abusing seminarians and minors, became the first US cardinal in history to resign.
Vigano, 77, was the Holy See's apostolic nuncio, or ambassador, in Washington from 2011 until 2016.
He has been a lightning rod within the Vatican who lost a power struggle in Rome under Benedict, emerged as a Francis critic, and reportedly ordered the halt of an investigation into the alleged sexual relations between an archbishop in Minnesota and seminarians.
Jason Berry, who has written several investigative books about the Vatican, said he believes this is the first time a pope has been accused from within.
Berry and his co-author Gerald Renner wrote Vows of Silence' about Catholic cover-ups, including a dramatic one about abuser Marcial Maciel, who founded the religious community Legion of Christ.
"Our book lays the cover-up question right at (Pope) John Paul II's doorstep. But from within the Vatican hierarchy, from within the Roman Curia, I don't think anyone has ever publicly accused a pope of covering up for a sex abuser," Berry said. "That's why this is such a big deal."
Vigano's letter said that McCarrick had been privately sanctioned under Benedict - though only after years of warnings about his alleged behaviour. The warnings that Vigano describes dealt with McCarrick's alleged behaviour toward seminarians and young priests - not toward minors. Viganò wrote that the measures, taken "in 2009 or 2010," banned McCarrick from travelling, holding Mass, or participating in public meetings.
Yet McCarrick appears to have done essentially the opposite. He regularly appeared as a speaker and celebrant at church functions and represented the church in prominent foreign diplomatic efforts in places like China and Iran.
A video from 2013 shows Benedict warmly greeting McCarrick in Rome, at the Pope's resignation (and the subsequent election of the new pope), where McCarrick gave round-the-clock television interviews and stayed at a seminary.
Vigano's letter also says that in 2013, he met Francis months into his papacy and told him face to face that there was "a dossier this thick" about McCarrick. He says he then told Francis about Benedict's order that McCarrick remove himself from public life.
"He corrupted generations of seminarians and priests and Pope Benedict ordered him to withdraw to a life of prayer and penance," Vigano says he told Francis.
"The Pope did not make the slightest comment about those very grave words of mine and did not show any expression of surprise on his face, as if he had already known the matter for some time, and he immediately changed the subject."
Vigano also alleges in that conversation that Francis told him American bishops "must not be ideologised, they must not be right-wing ... and they must not be left-wing, and when I say left-wing I mean homosexual."
It was not possible to reach Benedict or his representatives right away. Francis has not commented previously about what he was ever told about McCarrick, and Vatican spokesman Greg Burke did not respond to a request seeking comment.
The American Catholic Church is deeply divided over Francis' leadership, with fault lines similar to those seen in the political realm. Francis' comments and teachings about everything from immigration and global warming to the death penalty are frequently adopted or refuted along partisan lines.
The Viganò document uses American culture-war language, such as "right-wing" and "left-wing," and concludes the letter by blaming "homosexual networks" for sexual abuse and corruption.
US conservative Catholics who have suspected Francis of surreptitiously opening the door for liberalising changes around sex and marriage have in recent years focused on the increased acceptance of LGBT people.
Common targets for right-wing blogs like LifeSite and ChurchMilitant are bishops and cardinals they deem too moderate or liberal. Constantly on this list is Washington's Donald Wuerl, Chicago's Blasé Cupich and Joe Tobin of Newark. All are named by Vigano as being linked by "wickedness."
In the letter, Viganò described several figures who could corroborate parts of his account. Those people could not be immediately reached.
