Varadkar took to Twitter on Friday to express his support to India, while adding that he was invited by Prime Minister Modi to visit India.
Prime Minister Modi also conveyed his wishes to Varadkar via Twitter.
Spoke to Ireland's new Taoiseach Leo Varadkar on phone to congratulate him and reaffirm our commitment for stronger India-Ireland ties.
— Narendra Modi (@narendramodi) June 16, 2017
38-year old Leo Varadkar was formally elected Taoiseach - Prime Minister of Ireland- at a confirmation ceremony in the Dail in Dublin after he won the Fine Gael party leadership earlier this month.
Varadkar will succeed Enda Kenny, who stepped down in May, following him involvement in a corruption scandal.
Varadkar, born in Dublin in 1979, is the son of an Irish Catholic nurse from County Waterford and a Hindu doctor from Mumbai.
His parents met in England in the 1960s and lived in India for a time before moving to Ireland, according to The Guardian.
As the next leader, he faces a number of challenges as he takes over the only EU country that has a land border with the U.K. as it prepares for Brexit.
Abortion will be another big issue for Varadkar, specifically demands for a referendum on the future of the eighth amendment of Ireland's constitution.
In his first speech as Taoiseach, Varadkar told the Dail, the Irish Parliament, "The U.K. has been through some terrible tragedies over the past few weeks so we offer them our sympathy and solidarity."
He vowed to approach the job as Taoiseach with "profound humility and respect for what has gone before".
Varadkar becomes the EU's youngest national leader with Jüri Ratas, the 38-year-old Estonian Prime Minister and Emmanuel Macron, 39 of France, joining him as a trio under 40.