The two bishops added in a statement: "Again we apologise to them".
More than 170 cases of abuse have been found to have been covered up by Dublin church leaders. Incidents began to be revealed in 1995 but many records were hidden until 2004.
The cases came to light after Andrew Madden, a former alter boy, made public the abuse he suffered at the hands of priests and the church's attempts to buy his silence in the mid-1990s.
Walsh and Field, who worked in the Dublin archdiocese, said that they had informed Archbishop Diarmuid Martin of their resignation over the issue.
Both said that the report had said that they had done nothing wrong.
On Wednesday, Bishop Jim Moriaty of Kildare resigned saying that he had allowed the "prevailing culture" of abuse against children to go unchallenged.
Bishop Donal Murray became the first bishop to resign last week following the report's publication.
Last week Bishop Donal Murray of Limerik resigned following the report.
Martin Drennan of Galway, a fifth Bishop named in the investigation, said that he did nothing to endanger children and will not resign.
The four other bishops initially took this line.
Madden said: "Martin Drennan will have to resign. We want full accountability".
The report, published on November 26, said that Church leaders in in Ireland had allowed child sexual abuse by priests to take place on a significant scale over a period of 30 years.
It said bishops had "obsessively" hidden abuse to protect the reputation of the Irish Catholic church from 1974 to 2004.