Update 7.51pm: The Labour Party leader, Brendan Howlin, has said Leo Varadkar must unite everyone in the country if he is to be elected as Taoiseach.
Mr Varadkar can be officially installed as Taoiseach as early as Tuesday week, June 13, and Mr Howlin streesed that Fine Gael's new leader must "deliver progress to all, and not just to a fortunate few".
Mr Howlin said: "On behalf of the Labour Party, I would like to extend my personal congratulations to Leo Varadkar on his election as leader of Fine Gael, today is a day that I’m sure will give enormous pride to him and to his family and friends. I would also like to congratulate Simon Coveney on a dignified and passionate campaign - I have no doubt that he will continue to play a significant role in the Fine Gael party for many years to come.
"If Leo wishes to command the support of Dáil Éireann to be elected as Taoiseach, I believe there are a number of changes to the Government of Ireland which he should commit to immediately. He will now have about 10 days to identify his priorities before his potential election as Taoiseach, and I believe he should use that time to set out his commitment to making substantial changes to how Ireland is currently governed."
The Labour leader called for the Dáil to get back to work "after a year of dormancy".
He said: "Those changes should include a genuine commitment to respect the decisions on the Dáil, and a set of policy commitments for which clear public support has emerged - for immediate changes to the leadership of an Garda Síochána; for a referendum to repeal the eighth amendment; for ramping up investment in infrastructure; for funding of a new model of healthcare; and for a renewed charter of basic rights for all in work.
"During the election campaign over recent weeks, we heard some language from Leo Varadkar that was clearly designed to divide rather than unite our people. To serve our country as Taoiseach, such statements will need to be consigned to the dustbin, and he will need to present a vision for our future that can unite all of our people and deliver progress to all, and not just to a fortunate few."
Earlier: Enda Kenny has paid tribute to Leo Varadkar on his election as Fine Gael leader.
He said: "This is a tremendous honour for him and I know he will devote his life to improving the lives of people across our country. He will have my full support in the work that lies ahead."
Mr Kenny also thanked Simon Coveney for making the Leadership election "a real contest".
Sinn Féin President Gerry Adams also congratulated Mr Varadkar, but warned that Fine Gael under his leadership will be dragged further to the right.
Mr Adams said that ordinary people will suffer as a result.
He said: "I congratulate Leo Varadkar on his appointment as Fine Gael Leader. However his elevation to the position of Taoiseach is another matter entirely.
"For a party that has been preaching democracy to other parties for three weeks now the nature of this election has been very undemocratic. Some 65% of the membership voted for Leo Varadkar’s opponent yet he managed to come through as the winner.
"Leo Varadkar is a Tory. Fine Gael will be dragged even further to right under his leadership and that can only mean further hardship for ordinary people.
"Leo Varadkar as Taoiseach will mean greater disadvantage. It will mean more people locked out of the housing market. It will mean the growth of precarious work and poverty wages."
The Sinn Féin President claimed it will mean a "widening of the gap between the rich and the poor".
He said: "It is now over to Micheál Martin to see if he will become the first leader of Fianna Fáil to elect two Fine Gael Taoisigh.
"The fact is that Leo Varadkar has no mandate to lead this state. In fact, he barely has a mandate to lead his party but that is up to them. He represents only a minority of Fine Gael.
"He should call a general election immediately and seek a mandate from the people.
"Minister Varadkar’s expressed position in relation to Designated Special Status for the North is very positive and I hope he will now move to make this his party’s position and carry it into an election."