Fianna Fáil abstained on the vote on Mr Varadkar, which was in accordance with the party's "confidence and supply" arrangement to support the minority Fine Gael government. "The government I lead will not be one of left or right, it will be one of a new European center where everyone gets a fair go".
"As our youngest taoiseach, he represents a modern, diverse and inclusive Ireland and speaks for them like no other", Kenny said.
In a move created to heal any divisions in his own party, Varadkar on Tuesday appointed the man he defeated, Simon Coveney, as deputy leader of Fine Gael.
Leo Varadkar is set to officially become the Republic of Ireland's Taoiseach (prime minister) on Wednesday.
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Mr Kenny will submit his letter of resignation to President Michael D Higgins at Áras an Uachtaráin this afternoon.
Enda Kenny came to power at a time of great uncertainty for the Republic of Ireland and Europe as a whole.
He surprised party members with the announcement that his leadership rival, Simon Coveney, had been appointed deputy leader of the party.
A qualified doctor who has held three cabinet posts, Varadkar only revealed he was gay months before Ireland became the first country in the world to back same-sex marriage in a referendum in May 2015.
The nomination was seconded by Dublin Rathdown TD Josepha Madigan, who said Mr Varadkar was "a leader for Ireland of today: open, decent and compassionate".
In a coming-out speech he gave in a radio interview, he had said, "It's not something that defines me".
The son of an Irish nurse and an Indian doctor is also due to announce his cabinet, following the vote, with Public Expenditure Minister Paschal Donohoe expected to be promoted to finance minister, replacing the retiring Michael Noonan.
His family originates from Varad in Maharashta's Sindhudurg district, and Leo Varadkar kept his Indian connect alive, completing an internship at the King Edward Memorial Hospital in Mumbai. "We wish him the very best", Prendergast said.
However it is his policies that will attract more scrutiny at home with opponents warning that the former health, tourism and social protection minister, who first joined the centre-right party aged 17, would nudge it further to the right.