He was ordained a priest in the capital in 1961.
During his years in the Dublin Archdiocese Fr Moriarty served as a chaplain in University College Dublin and as parish priest in the suburbs of Deansrath and Donaghmede.
Thirty years after his ordination, the clergyman was made Auxiliary Bishop of Dublin.
It was in this role that the bishop became central to the so-called Fr Edmondus case which exposed the extent of the Catholic Church's power in 1960s Ireland.
Bishop Moriarty was asked to look into and discuss child abuse allegations against the cleric with local priests and then Archbishop, and since retired Cardinal Desmond Connell.
But the Murphy report found the investigation was not taken any further.
In the wake of the damning probe last November Bishop Moriarty - appointed Bishop of Kildare and Leighlin in 2002 - admitted Church authorities had failed to respond properly to child abuse for decades.
"I fully accept the overall conclusion of the Commission that the attempts by Church authorities to protect the Church and to avoid scandal had the most dreadful consequences for children and were deeply wrong," he told mass-goers.
"This is a dark and dispiriting chapter which has scandalised and shamed our Church and I wish to apologise to all who have been hurt."
He initially refused to tender his resignation insisting he had not been singled out personally for cover-ups or mishandling.
But two days before Christmas last year, Bishop Moriarty said he hoped his resignation "honours the truth that the survivors have so bravely uncovered and opens the way to a better future for all concerned".
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