Augustinian Fr. Gabriel Daly criticized the absence of Irish bishops at the funeral mass of Marist Fr. Sean Fagan, who was censured by the Vatican in 2008.
Redemptorist Fr. Gerry Moloney told NCR "something is rotten in the state of the CDF, and while the current people and processes remain in place, nothing will change."
Analysis: The pope may have exposed church tensions over how to engage the gay community -- tensions that have intensified after the Orlando mass shooting at a gay nightclub.
German Cardinal Reinhard Marx also told NCR that Pope Francis' Vatican reforms depend on "a new relationship between lay people and clerics, that is clear."
Archbishop Diarmuid Martin confirmed that Pope Francis, or his successor, will visit Ireland in 2018 for the World Meeting of Families.
One of the issues that is likely to be to the fore in Ireland's general election, slated for Feb. 26, is the future of denominational education. The Labour Party has pledged to end the "baptism barrier" to school entry if re-elected.
The Association of Catholic Priests in Ireland is to write to the Vatican's Congregation for Bishops expressing its members' "grave disquiet" over the current selection process for Irish bishops.
Augustinian Fr. Gabriel Daly said that the 1968 papal encyclical had never been accepted by the church at large. Daly also condemned the church over its treatment of women.
"Throughout the debate and the discussion, we did ask people to try to be respectful and inoffensive in language," Archbishop Eamon Martin said.
Catholic Ireland has become the first country to introduce gay marriage by popular vote, with 62 percent voting "Yes" in a referendum on Friday.