"The public have spoken".
Mr Varadkar said the results represented "the culmination of a quiet revolution", one that had been taking place in Ireland for the past 10 to 20 years.
Defying all expectations of a close contest, exit polls in Ireland predicted that the nation voted overwhelmingly on Friday in favor of repealing a near-total ban on abortion in a referendum.
He added that Irish voters "trust and respect women to make the right choices and decisions about their own healthcare".
After the "Yes" vote is confirmed in official results, the eighth amendment of Ireland's constitution will be repealed.
As a result, Irish women would wished to terminate their pregnancies had to seek illegal options within the country - at the risk of being thrown in jail for 14 years - or travel overseas.
The referendum followed months of debate between "yes" and "no" campaigners on whether the Irish Constitution's Eighth Amendment - which gives the mother and fetus an "equal right to life" - should be scrapped.
Its repeal will mark a significant victory for women's rights in a country that only legalised divorce in 1995. However, a wrong does not become right simply because a majority support it... "A sigh of relief that the Ireland we always thought we had, has now been recognised".
The country's largest newspaper, the Irish Independent described the result as "a massive moment in Ireland's social history".
The law was tweaked in 2013 to allow terminations if the mother's life is at risk.
Voters backed the removal of a wide-ranging constitutional block on terminations by about 68 percent to 32 percent, according to exit poll of about 4,500 voters commissioned by the Irish Times.
John McGuirk, communications director for the "Save the 8th" campaign pushing a "No" vote, reacted to the exit poll on Twitter. "We just can't believe that in the 21st century, something like this would happen".
Draft legislation released before the referendum would allow for relatively unrestricted abortions up until 12 weeks of pregnancy, subject to consultation with a doctor and a short waiting period. The unborn child no longer has a right to life recognised by the Irish state.
The Lib Dem leader said: "Since there is, effectively, direct rule from Westminster, the government has responsibility and it can and should take the opportunity to deal with this issue properly".