Approximately 800 infants aged between 35 weeks and three years old are thought to be buried in the mass grave at the site of the former Bon Secours Mother and Baby Home in Tuam, County Galway by a commission set up to investigate alleged abuse at such homes.
The Commission of Investigation into Mother and Baby Homes found that most of the remains were buried in the 1950s.
The home, which was run by the Bon Secour order of Catholic nuns, closed in 1961.
In a letter to Ireland's attorney-general, the 63 members of Irish First Mothers allege that they suffered "religiously motivated grievous injuries" at the hands of those charged with taking care of them at similar institutions across the country.
"With respect to genocide, it is the religious mind-set of the perpetrator (not the victim) which is pertinent," the letter claims.
"We assert that the perpetrators were malignly motivated by their own Catholic ideological characterisation of us as a religiously defined group: a caste of so-called "fallen women"."
"Irish society has a historic, deep Catholic veneration of the "virgin mother" as a deity figure.
"Thus us unmarried mothers were automatically deemed offensively faithless; viewed culturally by perpetrators as bereft of rights."
In the letter, the women also claim that they suffered "lifelong psychological injuries" because they were forcibly separated from their children.
Catholic Archbishop Michael Neary of Tuam said in a homily on Sunday that he was "greatly shocked" to learn of the extent of the numbers of children buried in the graveyard at the home.
He stressed that the Archdiocese had nothing to do with the running of the institution and that any useful material the Archdiocese possessed has already been handed over to the Commission and will continue to assist investigators.
Archbishop Neary added: "Those who have suffered are uppermost in our minds and at the very heart of our prayers."