Fine Gael MEP Brian Hayes has called on the Greek political leaders to "ditch their aggressive, provocative language".
In a statement released before the official result was announced but with polls showing a No vote ahead, Mr Hayes said the result of the Greek referendum must be respected by European authorities and leaders.
“I think there is now an urgent need to resume negotiations in a spirit of good faith,” he said.
Mr Hayes said the most immediate issue was funding for the Greek banks.
“Ten days ago the two sides were very close to an agreement. The Greek government has to be clear in its own mind if it wishes to remain in the euro zone.
“There are elements in the Greek government who wish to leave the euro. It is critical that the Greek government now make an application for a third bailout in seeking almost €30 billion of new funding.”
Mr Hayes said funding could not be provided without conditions attached.
“For the sake of the Greek people it is important that negotiations proceed with a sense of urgency. If negotiations are to proceed in an atmosphere of compromise it is also necessary that Greek political leaders ditch their aggressive, provocative language.
"Every government in Europe has a democratic mandate. The Greek government is not special in this regard."
He said confidence in the currency would only be maintained by the fair application of agreed rules and procedures.
Minister for Finance Michael Noonan said on Sunday night he hoped discussions between the Greek Government and other European states would resume.
“I hope that following this result the Greek Government will continue discussions with fellow member states in an effort to provide certainty for the Greek people and return stability to their economy,”Mr Noonan said.
"Ireland will continue to engage in an effort to reach a successful conclusion to the negotiations".
Meanwhile, Sinn Féin leader Gerry Adams has said he commended the people of Greece for their "historic decision" to take a stand against austerity and an unsustainable bailout package.
"This democratic decision must be respected by all EU leaders," Mr Adams said.
The Sinn Féin leader described the referendum as a significant exercise in democracy on behalf of Greece and the rest of the EU.
"I also want to comment Prime Minister Alexis Tsipras and his Government for the courageous leadership they have shown.
"The big question facing all citizens of the European Union today, including Irish citizens, is whether people have the right to assert democratic control over decisions which affect their lives. That is at the heart of the crisis in Greece. It is also a huge issue in Ireland.
“The EU must now make clear its desire to keep Greece in the euro zone by negotiating a solution that doesn’t impose years of economic depression on the Greek people.”
Fianna Fáil Leader Micheál Martin has called on the Irish government to play a positive and constructive role in the political process that would follow the Greek referendum outcome.
“Now that the public vote in Greece has concluded we express our hope that the situation in Greece can move on in a more constructive way whether this is possible within a European framework is hard to tell following the referendum result,” he said.
Mr Martin said all Europeans should be conscious of the appalling hardship endured by the people of Greece.
“The chaos that we have seen in the last week has not done anything positive for Greece’s prospects or for European stability.
“It is far from clear what the outcome of this referendum means in real terms, but that clarity is needed and needed urgently.”
Mr Martin said the European Union was better and stronger when member states were united in common purpose, “and debt sustainability plays a key part in this”.
He added: “We would hope also that the Irish government might play a more positive and constructive role in the political process that follows now than it has to date”.
