They want Bishop Martin Drennan, Bishop Ray Field, Bishop Jim Moriarty, and Bishop Eamonn Walsh to give up their school positions.
The Irish Primary Principals Network (IPPN) also wants the four bishops to be accountable for their actions -- or inactions -- in discharging their child protection responsibilities.
It is an unprecedented intervention by primary school heads -- and their first public statement since the publication of the damning Murphy report into the church's handling of complaints about paedophile priests in Dublin.
"This is not a 'witch hunt'; it is about justice and closure for all those whose dignity has been damaged beyond the most basic norms of any society in any era," said IPPN director Sean Cottrell.
In an IPPN survey of 630 principals, four out of five agreed that the bishops named in the report should not continue in their position as patrons.
In all, around 3,000 primary schools are under the patronage of Catholic bishops. All teacher appointments to Catholic schools have to be approved by them and they set up selection boards to appoint principals.
Under the Education Act, the Education Minister has responsibility for registering school patrons but the act does not set out procedures for removing a patron from the register.
The IPPN survey also found that 2pc of schools admitted they did not have up-to-date child protection policies -- a situation previously confirmed in a Dail reply to Fianna Fail TD Mary O'Rourke who was told that in September a total of 266 schools had still not availed of continuing professional development to support the implementation of the Stay Safe programme.
That figure had since been cut to less than 200, said Education Minister Batt O'Keeffe.
Shock
However, Mr Cottrell expressed shock that some schools had still not implemented the programme. Child protection guidelines should be put on a statutory footing, he added.
"Regardless of circumstances, anyone whose actions or in-action as the case may be, who has failed to protect even a solitary child, must be investigated and held to account by the judicial system," he said.
But the INTO last night moved to assure parents that primary schools strived for the highest standards of child protection, saying there was clear evidence of this in school inspection reports.
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