Varadkar will be voted in as Ireland's 14th prime minister by parliament on Wednesday, having secured the support of the independent lawmakers who back the Fine Gael-led minority and an agreement by the main opposition party Fianna Fail to abstain.
At 38 years of age, the openly gay Varadkar - the son of an Irish nurse and a doctor from India - becomes the youngest prime minister since independence from Britain in 1922.
"I've been elected to lead but I promise to serve", Varadkar said, according to BBC News.
The government I lead will not be one of left or right.
Enda Kenny has said he hopes his legacy as the Republic's Taoiseach will be marked by a modest contribution to improving the country.
Varadkar later met President Michael D Higgins, who gave him the seals of office to officially confirm his appointment.
It's believed Agriculture Minister Michael Creed was the first to get the call from the newly appointed leader.
In his first speech, Varadkar said the United Kingdom has been through some awful tragedies over the past few weeks so we offer them our sympathy and solidarity.
As he moves forward as prime minister, he will face challenges including addressing Brexit, monitoring his country's relationship with Northern Ireland (which is part of the UK), and pushing government to run effectively.
Mr Kenny is also the only Fine Gael leader to be re-elected taoiseach. "In the months ahead, I will seek to do exactly that".
While analysts expect few major policy shifts from Varadkar and his new cabinet, the new Fine Gael leader has hinted at some changes.
Adams said earlier that his party would vote against him noting that "not all young people are progressive" and expressing fears the new taoiseach would "drag this government even further to the right".