Taoiseach Leo Varadkar has arranged to meet with Fianna Fáil leader Micheál Martin next week to "discuss ways forward" on forming a government.
Fine Gael now leads a minority government propped up in a confidence-and-supply arrangement with republican party Fianna Fáil.
He handed over his resignation to President Michael Higgins after a tempestuous but inconclusive sitting of the Dáil Éireann, which met for the first time since the February 8 election.
'In accordance with the constitution, the [prime minister] and the government will continue to carry on their duties until successors have been appointed'.
Fine Gael, Mr Varadkar's party, came third after Fianna Fail and Sinn Fein.
Sinn Fein's Mary Lou McDonald secured the most support beyond her own lawmakers in the first ballot for prime minister, adding eight left-wing votes to her 37 Sinn Fein deputies, which left her far short of the 80 required to form a government.
After a 2016 vote it took 70 days for a minority coalition government to form under Fine Gael, which was supported in a confidence and supply agreement by Fianna Fail.
"You voted for a government for change and that is what Sinn Féin is determined to deliver". He could potentially end up leading the country if he can gain the support of Fine Gael.
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"Essentially, it means that Ireland will contribute much more to the European Union budget, but will actually receive less back, in terms of payments to Irish farmers, and also funds for regional development and social development", he highlighted.
McDonald, who has said a Fianna Fail-Fine Gael coalition would be "a slap in the face to the Irish electorate", hoped for a change of mind from Fianna Fail on her party, although there is so far little sign of that.
Sinn Fein, which picked up 14 seats, attracted voters by focusing on Ireland's housing shortage and health crisis.
Sinn Fein's negotiating team, which includes Pearse Doherty, Eoin O Broin, Louise O'Reilly and Matt Carthy, have held talks with the Green Party, Social Democrats, Rise-Solid- arity-People Before Profit and Independent TDs.
The IRA fought British rule in Northern Ireland for decades until the late 1990s, when a peace agreement largely ended violence that left more than 3,000 dead on all sides.
He wants a referendum on the union of the two parts of the island in five years.
He stipulated that this "could really only happen on the basis that there's full respect for the fact that we did win 450,000 votes in this election".
"I think it is a matter of great regret that Fianna Fail in particular continues to maintain a stance of excluding Sinn Fein and not just us", McDonald told reporters.