"What we see is the culmination of a quiet revolution that has been taking place in Ireland over the last couple of decades", Varadkar, who became Ireland's first openly gay prime minister a year ago, told journalists in Dublin.
"There is no prospect of the (abortion rights) legislation not being passed", McGuirk added.
"We are waking up to an Ireland where the Irish public have supported a woman's right to choose and a woman's right to make decisions in relation to her pregnancy".
Ireland's Minister for Children and Youth Affairs Katherine Zappone said she was grateful to the voters for repealing the eighth amendment.
Campaigners who have fought for more than three decades to remove the Eighth Amendment abortion ban from Ireland's Constitution hailed the referendum vote as a major breakthrough for the largely Catholic nation.
Earlier Saturday, a leading anti-abortion group called the vote a "tragedy of historic proportions" as it conceded defeat.
"They were women in crisis pregnancies or women in awful circumstances; victims of sexual violence or those who have been given terrible news regarding the viability of the baby in the womb, a baby they dearly want, or women whose lives are put at risk by an imminent childbirth".
Abortion is still banned in some 20 countries worldwide, while others have highly restrictive laws in place. Lawmakers are now expected to debate proposed legislation allowing abortions within the first 12 weeks of pregnancy, and after that in cases of fetal abnormalities or serious risks to the mother's health.
The Friday vote saw citizens effectively opt to either retain or repeal the Eighth Amendment of the country's Constitution, which prohibits terminations unless a mother's life is in danger.
The referendum will likely end the need for thousands of Irish women to travel overseas - mostly to neighbouring Britain - for abortions they can't get at home.
Younger voters showed more support for overturning the amendment than older voters, according to RTÉ exit poll projections.
The law on abortion is enshrined in the country's constitution, which can be changed only by referendum.
Catherine Claffey, 53, a flower seller on the capital's main Grafton Street shopping thoroughfare, said: "We've pulled ourselves out of the dark ages". The poll had a margin of error of plus or minus 1.6 percent.
The landslide vote, celebrated outside Parliament Saturday, shows an overwhelming desire for change in a country once so closely entwined with Catholic tradition.More news: Boba Fett Film Coming From James Mangold
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