In a tweet Mrs May congratulated Ireland over the referendum result.
He said it will be remembered as "the day Ireland stepped out from under the last of our shadows and into the light".
Children's Minister Katherine Zappone said she felt emotional and expressed deep gratitude to voters.
The group's leader John McGuirk added: "The unborn child no longer has a right to life recognised by the Irish state".
"I believe in the principle of the right to choose - I think in the rest of the United Kingdom we have a sensible balance", said Sir Vince.
"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 was first banned in Ireland back in 1861 by the Offences Against the Person act, and the ban remained in place after Irish independence.
An exit poll by the newspaper said that about 69% of the people had voted to repeal the constitution's eighth amendment that effectively bans terminations.
A spokesman for an anti-abortion umbrella group, Save The 8th, conceded there was "no prospect" the country's abortion ban, imposed in a 1983 referendum, would be retained.
Over 66 percent of roughly 2.1 million voters supported eliminating the nation's Eighth Amendment, election official Barry Ryan said Saturday, opening the door for Ireland's government to consider legislation permitting abortions during the first 12 weeks of pregnancy. The Irish Times newspaper predicted 68 percent of voters cast ballots to change the law in an exit poll published earlier.
Abortions are now only legal in Northern Ireland if the life or mental health of the mother is at risk. As a result, thousands of Irish women make the trip overseas, often to England, to have an abortion.
Deputy Prime Minister Simon Coveney said he believed a middle ground of around 40 percent of voters had decided en masse to allow women and doctors rather than lawmakers and lawyers to decide whether a termination was justified. Amnesty International Ireland and the Abortion Rights Campaign, for instance, were required to return political grants of $150,000 and $25,000, respectively, to George Soros's Open Society Foundations.
But Irish campaigners against the changing of the constitution have said they will continue to debate the issue, despite what some have called a "sad day" for Ireland.
The historic vote was on whether Irish women would be able to access abortions in the country, and whether the constitution should be changed to reflect this. The poll has a margin of error of plus or minus 1.6 percent.
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Landslide Irish vote repeals restrictive abortion amendment
Love them at every stage of life". "You may feel that the country has taken the wrong turn, is no longer a country you recognize. Majority Catholic Ireland is now in a heated debate over legalizing pregnancy termination, which is banned under current laws.