Vardkar, a first openly gay prime minister in waiting who is the son of a Mumbai-born father and Irish mother, is due to be formally voted in as Taoiseach, or Irish prime minister, next week.
UK Prime Minister Theresa May's "failure to get a majority" showed there is no strong support for a hard Brexit, he said.
"The Irish government is ready to participate in negotiations on Brexit and to restore power-sharing in Northern Ireland. We must ensure that the Brexit talks are handled in a smooth and coherent manner to secure the best possible outcome for Ireland, for Europe and the UK.
He said another priority is the early restoration of the Stormont Executive in Northern Ireland, the devolved administration.
"There is now a strong opportunity for the parties in Northern Ireland to re-engage in discussions to form an executive," he said.
Varadkar became Ireland's Taoiseach designate last week after winning the leadership of the ruling Fine Gael following Enda Kenny's decision to step down after six years at the head of government in Dublin.
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