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Mr Varadkar is to contact Mr Coveney to discuss what position he will hold in government and the future of the party.
He is also expected to hold talks with Fine Gael's partners in government and with Fianna Fáil next week.
The former GP is the son of an Irish nurse and a doctor from India and much of the media coverage of his victory has focused on Mr Varadkar's background, age and sexuality.
The former first minister of Northern Ireland, DUP leader Arlene Foster, has spoken on the phone to congratulate Mr Varadkar. He has also received a letter from Prime Minister Theresa May.
'Prejudice has no hold'
Following the announcement on Friday evening, the new Fine Gael leader said he was "honoured" to accept the "enormous challenge" ahead of him with humility.
"If my election shows anything it's that prejudice has no hold in this Republic," he said.
"When my father travelled 5,000 miles to build a new home in Ireland, I doubt he ever dreamed his son would grow up to be its leader."
He said Simon Coveney had gained his and the party's admiration for his "principled and spirited" campaign and he looked forward to working with him to bring Fine Gael and Ireland forward.
"Any child growing up in Ireland now, I hope, looks at me and my unlikely story and my background, and everything about me, and perhaps says to themselves, that there is no office in this state that I can't aspire to, nothing that they can't do if they believe in themselves," he said.
"But our job as a party is to make sure that every person in our country actually has those opportunities, because we don't have equality of opportunity in this country, there is great inequality of opportunity in this country, but as a party I want to dedicate ourselves to building a republic of opportunity."
He said restoring devolution in Northern Ireland and dealing with Brexit would be immediate priorities.
"Northern Ireland will leave the European Union with the United Kingdom, but we do think there should be special arrangements for Northern Ireland," he said.
"It is a special place and we would like to work with the new executive when it is formed to define what that is.
"But particularly, what we'd like to see is there to remain as close a relationship as possible between the north and south of Ireland and an invisible or non-existent economic border in particular and Northern Ireland remaining in as many EU programmes as possible."
Mr Varadkar also paid tribute to Mr Kenny for passing on a party and country "in a much better place than he found it".
The leadership was decided in an electoral college system that gave 65% of the vote to the Fine Gael parliamentary party - made up of 73 TDs (members of the Irish assembly), senators and MEPs.
The party's 21,000 rank-and-file members had 25% of the vote, and 235 local representatives had 10%.
Mr Varadkar, who came out as gay in the run up to the 2015 same-sex marriage referendum, has been in charge of the welfare system.
He has come to personify the liberalisation of a country which was once regarded as one of Europe's most socially conservative nations - homosexuality was illegal until 1993.
However, Mr Varadkar has come under criticism for his comments on progressive issues and workers' rights.
The new Fine Gael leader will have to be endorsed by the independent members of the minority coalition government before he becomes taoiseach. That's expected to happen later this month.