Vigano's claims have raised speculation of a campaign against the pontiff by conservatives in the Church.
But, he writes, the pontiff ignored that, and allowed McCarrick to continue to publicly serve the church for another five years.
Vigano said he had told Francis in June 2013, just after he was elected pope by his fellow cardinals, about the accusations against McCarrick.
Cardinal Donald Wuerl, Archbishop of Washington, enters church for Mass at St. Mathews Cathedral, Wednesday, August 15, 2018 in Washington.
McCarrick was forced to leave his seminary and live a life of penance after former Vatican ambassadors in Washington, now dead, reported him for "gravely immoral" behaviour with seminarians and priests.
The archbishop told CBS News he was speaking out "to combat the grave situation in the church, to protect the church and also to stop future abuse".
The Vatican's ambassador to the US from 2011 to 2016, Vigano said his two immediate predecessors "did not fail" to inform the Holy See about accusations against McCarrick, starting in 2000.
With LGBTQ families and the church's refusal to allow gay Catholics to participate in the Irish gathering, one reporter asked the Pope what he would say to a parent who "observe homosexual traits" in their child.
Vigano in the letter claims that he told Francis in 2013 about "a dossier this thick" with information about McCarrick's alleged misconduct toward young members of the church.
During a Q&A with journalists on his flight back to Vatican City from Ireland, the Pope said, "Don't condemn". On Monday, Zappone released her remarks to Francis and a letter she gave him laying out background for the monetary appeal.
Viganò contends that McCarrick's successor was "obviously, the first to have been informed of the measures taken by Pope Benedict". And, this is why I wanted to repeat it in the speech... and it was not at the airport, I was mistaken, it was at the president's. Responding to a plea from the adoptees, the pope assured the aging mothers it wasn't a sin to look for the children taken from them so long ago.
"May the Lord keep this state of shame and compunction and give us strength so this never happens again, and that there is justice", Francis said. He said he spoke directly to Cardinal Wuerl on several occasions and that Cardinal Wuerl "failed to acknowledge receipt of my two letters, contrary to what he customarily did".
His eleven-page letter accused senior Vatican officials of knowing as early as 2000 about the former archbishop of Washington, ex-Cardinal Theodore McCarrick.
Cardinal McCarrick became the first cardinal in living memory to resign his position in the Church leadership after a review concluded that claims he had sexually abused a 16-year-old boy were credible.
The official accused the pope of having known of allegations of sex abuse by a prominent US cardinal for years. "Further, the confidential settlements involving acts by Archbishop McCarrick in the Diocese of Metuchen and the Archdiocese of Newark were not known previously to Cardinal Wuerl or the Archdiocese".
He said on Saturday that the corruption and cover-up of abuse amounted to human excrement, according to victims. At least seven had ties to Illinois, prompting Illinois Attorney General Lisa Madigan to seek a complete accounting of all sex abuse allegations against those priests.