The referendum repeals Ireland's Eighth Amendment, which was added to the constitution in 1983, which banned abortion in Ireland unless there was a "real and substantial risk" to the mother's life.
Speaking ahead of the official result on Saturday afternoon, Varadkar said the success of the campaign was down to a number of "very fearless women and men" who told their stories and their experiences of how this "hard law created so many hard cases".
Ireland's push to liberalise its laws is in contrast to another traditionally Catholic European country, Poland, where the ruling conservative party and still powerful church are seeking to ban most abortions. 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.
Calling the result a culmination of a "quiet revolution" that had been gaining strength in the last 20 years, Varadkar said the large margin of victory will give his government a greater mandate when enacting new abortion legislation through parliament. "A quiet revolution has taken place", Varadkar, who became Ireland's first openly gay prime minister past year, said in a speech after the vote.
Varadkar said he'd like to pass legislation for a new abortion law by the end of the year, according to The Guardian.
"What Irish voters did yesterday is a tragedy of historic proportions", the Save The 8th group said.
Many lawmakers who campaigned for a "No" vote said they would not try to block the bill.
Saturday's result removes the equal right to life of the unborn and mother from the constitution.
International Development Secretary Penny Mordaunt said Ireland's vote was a "hopeful" day for Northern Ireland.
Its repeal will mark a significant victory for women's rights in Ireland.
Many people left floral tributes and handwritten notes saying that her death had influenced their decision to vote.
"Wonderful, wonderful, today is wonderful!" said 65-year-old Eileen Shields, who had been ostracised for falling pregnant outside of marriage when she was 18.
Northern Ireland's elected assembly has the right to bring its abortion laws in line with the rest of Britain, but voted against doing so in February 2016 and the assembly has not sat since the devolved government collapsed in January 2017.
But the Prime Minister faces a political headache over calls to act because her fragile administration depends on the support of the ten Democratic Unionist Party MPs - who strongly oppose any reform to Northern Ireland's strict laws.
"The polls suggest all generations voted with us", Catherine Conlon, a Trinity College professor told ABC News, after exit polls showed overwhelming support for repeal of the amendment. Abortions approved by doctors are allowed in the rest of the United Kingdom until the 24th week of pregnancy, but not in Northern Ireland.
He continued: "We have one last request, that the new law, that it is called 'Savita's law.' It should be named for her".