A source close to Vigano - who retired in 2016 at age 75 - confirmed the letter is genuine.
"I really was taken aback by, not just his words, but the derisive language and scorn behind them because that wasn't anything close to anything I'd ever experienced with him", Cupich said.
"You have the sufficient journalistic capacity to draw conclusions", the pontiff told reporters in the Vatican's plane.
"I am your father, I am your mother, let's talk, I will not throw you out of the family". Cupich said he hosted Vigano three times in Chicago: once for when he was installed as archbishop and two other times.
88-year-old McCarrick resigned last month in disgrace and was stripped of his title after allegations that he had abused a minor almost 50 years ago and also forced adult male seminarians to share his bed.
He said he would "not say one word" on the 11-page document, in which a former Vatican official, Archbishop Carlo Maria Vigano, says Francis should resign.
"In this extremely dramatic moment for the universal Church, his extremely dramatic moment for the universal Church, he must acknowledge his mistakes and, in keeping with the proclaimed principle of zero tolerance, Pope Francis must be the first to set a good example for cardinals and bishops who covered up McCarrick's abuses and resign along with all of them", Vigano said. In them, he begged not to be transferred to the Vatican embassy in Washington from the administration of the Vatican City State.
"He corrupted generations of seminarians and priests".
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McCarrick, now 88, resigned in disgrace in July.
"We ask forgiveness for the abuses in Ireland", said Pope Francis at the Mass, listing a litany of abuses including clerical sex crimes, a lack of compassion and action by church leaders and the separation of single mothers and children.
Earlier, he said no-one could fail to be moved by stories of those who "suffered abuse, were robbed of their innocence and left scarred by painful memories", and reiterated his wish to see justice served. 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.
Years of sexual abuse scandals have shattered the credibility of the church, which dominated Irish society four decades ago.
Hundreds of kilometres (miles) away, a few hundred sombre protesters marched through the Irish town of Tuam and recited the names of a 796 babies and young children who died at a Catholic-run orphanage there, most during the 1950s.
Pope Francis has declined to comment on a claim that he ignored sexual abuse allegations against a senior clergyman amid speculation conservative elements in the Catholic hierarchy are using the issue to mount a "putsch" to remove the liberal pontiff.
Last month, the Pope accepted McCarrick's resignation following the release of evidence that he had abused a child.
His trip was met with enthusiastic crowds but also protests, with about 5,000 abuse victims and supporters attending a "Stand for Truth" rally in the capital Dublin.
The Ireland that the Pope visited is a different country to that which greeted Pope John Paul II in 1979.