The pope instructed Baltimore Archbishop William Lori to conduct an investigation into Bishop Bransfield and appointed him as the apostolic administrator of the Diocese of Wheeling-Charleston in West Virginia.
DiNardo has said he wants Francis to authorize a full-fledged Vatican investigation into ex-Cardinal Theodore McCarrick, who was removed as cardinal in July after a credible accusation that he groped a teenager.
The USCCB headquarters in Washington, D.C., released a short statement more than three hours after the meeting saying the pope had "listened very deeply from the heart" as they spoke to him in "a lengthy, fruitful and good exchange" about how the USA church had been "lacerated by the evil of sexual abuse".
While Wuerl has argued parishioners should be allowed to know if a priest facing allegations of child molestation has been moved to another church, he allegedly engaged in the practice himself. The Vatican Press Office explained that, after hearing the Council of Cardinals, the Pope made a decision to convene the meeting, which will be tackling the theme of "Protection of minors".
Wuerl has said he plans to travel to Rome in the near future to meet with Pope Francis to discuss his resignation.
The Cardinal said the Pope had asked him to work out what he wanted to do "as we face new revelations of the extent of the horror of the clergy abuse of children and the failures in episcopal oversight".
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On the eve of the USA audience, Francis announced he was summoning the presidents of bishops conferences around the world to a February summit to discuss prevention measures and protection of minors and vulnerable adults.
An exhaustive, almost 900-page grand jury investigation into sexual abuse by Pennsylvania priests mentions Wuerl over 200 times.
Also involved are two officials from the conference, Los Angeles Archbishop Jose Gomez and Monsignor Brian Bransfield.
House Speaker Paul Ryan (R-WI) commented Thursday on the sexual abuse and cover-up scandal that is rocking the Catholic church.
Pope Francis accepted the resignation of West Virginia's bishop on Thursday and ordered an investigation into claims he sexually harassed adults.
He added: "Elitism, clericalism fosters every form of abuse".
Wuerl has faced calls to resign, including from a local deacon at St. Matthew's Cathedral and many other D.C. -area Catholics over his handling of sex abuse allegations by Catholic clergy in Pennsylvania when Wuerl was bishop there almost 30 years ago.