In an interview published Wednesday by Italian newspaper La Repubblica, Burke addressed Abp. Carlo Maria Viganò's call — echoed by a growing number of Catholics — for the Pope to step down.
"I cannot say it is wrong," he said, reinforcing the need for deeper inquiry. "I can only say that to arrive at this, one must investigate and respond in this regard."
"The request for resignation is in any case licit; anyone can make it in the face of whatever pastor that errs greatly in the fulfillment of his office, but the facts need to be verified," the cardinal added.
Last weekend, Abp. Viganò, former papal nuncio to the United States, issued an explosive statement alleging that Pope Francis knew of former Cdl. Theodore McCarrick's history of sexual predation and covered for him.
Burke described how Viganò's revelations had jarred him.
"I was deeply shaken because the entire document is most grave," he said. "I had to read it several times because the first reading left me speechless."
"I believe that at this point there is need for a complete and objective report on the part of the Pope and the Vatican," he added, reinforcing the need for deeper inquiry. "It is necessary that there is clarity, by going through all the documents to arrive at the truth."
Burke also echoed Viganò's warning that certain members of the hierarchy are working to change Church teaching on homosexuality.
"Yes, there are attempts to relativize the teaching of the Church according to which a homosexual act is intrinsically bad," he said, pointing specifically to the 2014 Synod of Bishops on the Family, "where the idea was presented that the Church should recognize the positive elements present in homosexual relations."
He also spotlighted the "support that Churchmen give to the Jesuit James Martin, who has an 'open' and wrong position on homosexuality," calling it "a problem."
Burke noted that "the data show that the major part of sexual abuse committed by priests are in reality homosexual acts committed with young people."
"I think a homosexual person cannot become a priest because he is not able to exercise in depth that paternity that is required. He must have all the characteristics to be a father," he said.
The cardinal's comments echo his statement immediately after Viganò's testimony was released.
"The declarations made by a prelate of the authority of Archbishop Carlo Maria Viganò must be totally taken to heart by those responsible in the Church," Cdl. Burke said last Sunday. "Each declaration must be subject to investigation, according to the Church's time-tried procedural law."
"After the truth of each declaration has been established," he continued, "then the appropriate sanctions must be applied both for the healing of the horrible wounds inflicted upon the Church and her members, and for the reparation of the grave scandal caused."