The remains of up to 800 babies and toddlers were found next to a home for unmarried mothers in Tuam, which was operated by the Bon Secours Sisters between 1925 to 1961.
A historian confirmed the names of the 796 children buried in the mass grave after she made repeated requests from the state for records.
The Tuam burial site was discovered in 1975 by 12-year-old friends Barry Sweeney and Francis Hopkins.
But, locally, it was referred to for years as a famine burial site where youngsters who had died in the 1840s disaster were buried in a mass grave, often on unconsecrated ground.
Records of hundreds more at other homes are still being held confidentially.
The discovery has led to calls for the Irish government to hold a short, focused public inquiry into the practices at so-called mother and baby homes over the last century, particularly mass burials.
A cross-department Government initiative will examine how best to address the Co Galway scandal.
A renewed focus on calls for Bon Secours nuns to fund a memorial headstone at the Tuam burial site has sparked massive interest nationally and internationally and led to calls for Taoiseach Enda Kenny to announce an inquiry and issue a formal apology to relatives of the dead.
Susan Lohan, co-founder of the Adoption Rights Alliance, said that the response so far is not good enough.
"I would be calling for these mass graves to be treated as if they were crime scenes. Talk about memorial stones at this point is way too premature. We should be inviting in forensics experts, forensic archaeologists."
The Archbishop of Tuam Michael Neary has said the archdiocese will co-operate with any inquiry - but added that it did not have any involvement in the running of the home.
"I was greatly shocked, as we all were, to learn of the magnitude of the numbers of children buried in the graveyard in Tuam. I was made aware of the extent of the situation by media reporting and historical research.
"I am horrified and saddened to hear of the large number of deceased children involved and this points to a time of great suffering and pain for the little ones and their mothers.
"I can only begin to imagine the huge emotional wrench which the mothers suffered in giving up their babies for adoption or by witnessing their death. Many of these young vulnerable women would already have been rejected by their families."
The Archbishop said that regardless of the time lapse involved "this is a matter of great public concern which ought to be acted upon urgently."
"As the diocese did not have any involvement in the running of the home in Tuam, we do not have any material relating to it in our archives. I understand that the material which the Bon Secours Sisters held, as managers of the Mother and Baby Home, was handed over to Galway County Council and the health authorities in 1961. "
He added that while Archdiocese of Tuam will cooperate fully, "there exists a clear moral imperative on the Bon Secours Sisters in this case to act upon their responsibilities in the interest of the common good."
"The Diocese will continue to work with the Sisters and the local community to provide a suitable commemorative prayer based memorial service, and plaque, and to ensure that the deceased and their families will never be forgotten.
"It will be a priority for me, in cooperation with the families of the deceased, to seek to obtain a dignified re-interment of the remains of the children in consecrated ground in Tuam."
However, Susan Lohan said that an unchristian attitude by the diocese at the time meant pregnant women ended up in such homes.
She claimed: "It was his diocesan parish priests who were probably belting the pulpit, and telling people that women and girls who had sex outside of marriage - and certainly those who became pregnant - had to be ostracised.
"So it is entirely down to their very unchristian practises of ostracising and despising these woman that they ended up in these homes."
On Thursday evening, a statement was released from The Sisters of Bon Secours religious order saying they were "shocked and deeply saddened" by recent reports.
"In 1961 the Home was closed. All records were returned to the local authority, and would now be within the Health Service Executive, Co. Galway," the statement read.
"The Bon Secours Sisters say they are committed to engaging with [the historian] Catherine Corless, the Graveyard Committee and the local residents as constructively as they can on the graves initiative connected with the site.
"The Sisters welcome the recent Government announcement to initiate an investigation, in an effort to establish the full truth of what happened."