He said he would make a "positive case" for the reunification of Ireland over time and within the EU.
"Everybody should have an equal opportunity to succeed".
"I would not have spent the last few years in politics if I was afraid of big challenges".
Earlier, the Social Protection Minister started his first day of campaigning at a 5k road race in Dublin.
An Indian-origin doctor and Ireland's first openly gay minister has emerged as the frontrunner in the Irish prime ministerial race.
He said Minister Simon Coveney is "a very fine politician" but said he believes Mr Varadkar has the "rare combination of intellect and judgement needed for leadership".
Supporters of Mr Varadkar say his straight-talking manner could widen the party's appeal after Mr Kenny's steady stewardship and boost their position in the polls ahead of elections likely late next year.
Mr Coveney said these included including some members of the Cabinet.
Coveney officially launched his campaign yesterday, saying that he would be a strong and stable leader of the party.
When Barack Obama became the United States president, such was his devotion to daily workouts that The New York Times called him "the exerciser in chief".
Fine Gael for the first time will be electing their leader through an electoral college which consists of the following; 73 members of the Parliamentary Party (PP) accounting for 65% of the total vote, nearly 21,000 party members accounting for 25% and 235 local representatives (232 councillors and 3 Údarás na Gaeltachta members) accounting for the remaining 10% of the vote.
"I'm really humbled, really grateful for that", he said.
"It would have to break two-to-one for Coveney among the party members to give him a chance, and only a chance".
Mr Varadkar was careful not to sound triumphant, however.
Setting out his stall, he said: "I want to take the country forward". Cork-born Mr Coveney (44), is the current housing minister, who has held the agriculture and defence portfolios.
Due to this surge in support from the parliamentary party for Mr Varadkar, Mr Coveney will need to secure more than 60% of the remaining votes from councillors and grassroots party members.