"The government I lead will not be one of left or right", he said, addressing the Dail.
Mr Varadkar has appointed his leadership rival Simon Coveney as deputy leader of Fine Gael.
Another unknown is how Ireland will work with the yet-to-be formalised coalition arrangement between the ruling Conservative party in London and the staunchly conservative Democratic Unionist Party of Northern Ireland, which opposes legal abortion and same-sex marriage.
As reported by the Reuters news agency, Varadkar's elevation marks another chapter in the social change that has swept through the country of 4.6m people that only decriminalised homosexuality in 1993 and legalised divorce two years later.
Mr. Varadkar, 38, will serve as Ireland's 14 taoiseach.
He replaces Enda Kenny, who stepped down as party leader that month and announced he would remain Taoiseach, or prime minister, until a successor was chosen.
He then travelled to the president's residence, Aras an Uachtarain, where President Michael D Higgins gave him the seals of office to officially confirm his appointment.
Leo Varadkar promised a "republic of opportunity" Wednesday after he became Ireland's youngest prime minister and the first who is openly gay.
He also said his father, who is an immigrant from India, would be proud that he could "be judged by his actions, not his origins or identity". Three years later, he was elected to represent the area in parliament.
Varadkar born in Dublin in 1979, is the son of an Irish Catholic nurse from County Waterford and a Hindu doctor from Mumbai.
Roscommon Fine Gael Senator Frank Feighan tweeted a picture of Varadkar making a speech, saying: "Copper face Jacks is rocking tonight!"
The first meeting of the new Government took place at Áras an Uachtarán last night with Mr Varadkar setting out his priorities such as Brexit, health and housing.
Naomi Long said: "Northern Ireland and the Republic have a good relationship and I am confident it will continue under his stewardship".