Tánaiste Micheál Martin has said he is "surprised" by the Taoiseach's resignation.
Speaking at Leinster House, Mr Varadkar said that being Taoiseach has been the most fulfilling time of his life, with Ireland becoming a more equal and modern place, including for women and LGBT people.
The leader of Fianna Fáil was speaking after the Taoiseach's announcement and said he and Green Party Leader Eamon Ryan were briefed by Mr Varadkar yesterday evening in relation to his plans and that he was surprised by his decision.
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Mr Martin said the three of them got on well, which was important in terms of the continuity and stability of the Government.
He wished Mr Varadkar well in both his personal life and career into the future.
"This is a coalition of three parties, not personalities, and I remain committed to the continuation of Government, to the fulfilment of our mandate and to the implementation of Government," Mr Martin said and added that there are still very serious issues to be dealt with.
Mr Martin said they would work with the newly elected Fine Gael leader in terms of continuing the Coalition.
He said the Taoiseach is entitled to take a decision of this kind for personal reasons and for political reasons.
"Our system of government as laid out in the Constitution is in terms of a Dáil majority, electing a government of the day and ministers of the day and that is no different as a result of this decision," he said.
Mr Martin said it is new and it is unprecedented in many ways but it has happened before where Taoisigh have been elected and mid-stream in different Dáil' but in his view, they have a clear mandate and a clear programme for Government.
Mr Martin said it was "not for any Fianna Fáil leader to comment on who the next leader of Fine Gael should be".
He said it was his sense that the public want politicians to focus on the day-to-day issues that concern them and that they have done a good job.
He said it was his view that governments should go full term.
"That creates stability, it avoids short-term decision making and it gives a better policy focus than a short-term electoral run," Mr Martin said.
"That is why I've been absolutely steadfast and consistent on this since the beginning of this Government that it should go full term and that remains my position," he added.
Green Party leader Eamon Ryan said Leo Varadkar "has been an energetic and committed leader of the country who was always supportive of his Government colleagues".
He thanked Mr Varadkar for his work as Taoiseach saying he had "served the country well and can be proud of the contribution he has made to Irish political life".
Mr Ryan said while the process of selecting a new Fine Gael leader and Taoiseach takes place, "the three coalition parties will continue to fulfil our mandate".
Speaking to RTÉ News, Mr Ryan said that while there are differences, he believes they have shown "real strength and solidity" as a Government.
He said he has been in previous governments where a Taoiseach stood down and while this Government is unusual with a rotating Taoiseach, he believes they can "continue the work of a stable Government".
Politics in Ireland has reached 'critical point' - McDonald
Irish politics has now reached a critical point in the aftermath of Mr Varadkar's announcement, the Sinn Féin leader has told the Dáil.
Mary Lou McDonald said the idea that Fine Gael would gather in conclave to choose a new Taoiseach without the input of the people is "unthinkable".
This a time for fresh leadership, not just a change of Taoiseach but a change of Government, she said.
Ms McDonald added that the Government has run out of steam and it is time to seek the dissolution of the Dáil and not to limp on.
Earlier, Social Democrats leader Holly Cairns said that while she was shocked when she heard the news of Mr Varadkar's resignation but it was not that surprising.
Also speaking to RTÉ News, Ms Cairns said the Government's record was one of failure on many issues and that Mr Varadkar could see that.
"I don't think his decision will instil confidence in the public, in the Coalition's ability to move forward and keep going," she said.
"I think it raises a lot of questions and I don't feel that a new Fine Gael Taoiseach is the answer to those issues affecting people every day. I think ultimately we need a new government, we need a general election," Ms Cairns added.
Labour Party leader Ivana Bacik said Mr Martin's admission of surprise at the Taoiseach's decision to step down "says it all" and that the move is "deeply destabilising".
Ms Bacik said it was "utterly untenable that we would see this Government continue".
She said the people should decide who leads the country and now was the time for an election.
"We need to see a change of Government, not a change of Taoiseach," she said.
Ms Bacik said: "What we've seen is Fine Gael really losing their reason to be in Government, and what we see now is a Government that has run out of steam, has run out of energy and has run out of ambition.
"Now is the time for a Government that is really stable, for a Government that has a really legitimate mandate, a Taoiseach that has a legitimate mandate to take over and take on these challenges and meet them."
People Before Profit TD Richard Boyd Barrett said he believes the people of Ireland will question their confidence in the Government.
Mr Boyd Barrett said people want a resolution to the things the Government have failed them on, such as housing and health.
"I think with this decision the Government should just accept that it is time for the people to decide who will be the next Government and give them the option to have a different approach to solve the problems that people really care about in this country," he said.
Mr Boyd Barrett said the decision to step down suggests a Government that is not confident in its ability to convince people that it can solve such problems.
