More than 60 percent of Greek voters voted against the conditions of the country's latest bailout, rejecting further austerity after years of cuts following the financial crisis.
European leaders were on Monday scrambling for a response, which could send Greece crashing outof the eurozone.
German Chancellor Angela Merkel was to meet with French leader Francois Hollande in Paris.
The pair spoke by telephone late on Sunday, declaring the decision must “be respected” and calling for an emergency eurozone summit which EU president Donald Tusk said would be held on Tuesday.
Renzi, who will meet Italian Economy Minister Pier Carlo Padoan on Monday morning, said he would make it clear to France and Germany that the format cannot just be dictated by the two countries, and requires the involvement of all leaders and EU institutions.
Meanwhile, German and French finance ministers were set for talks beginning in Warsaw at 8.00 GMT, while the Euro Working Group of top treasury officials will meet in Brussels.
Italian President Sergio Mattarella on Sunday urged Europeans and EU leaders to show solidarity with Greece and respect the outcome of the referendum.
“Greece is part of Europe, and the other people of the European Union owe its people their solidarity. This will in the coming days be the principle guiding Italy's actions and I hope this will be true also for the representatives of the Greek people, other European partners and the EU's institutions,” he said in a statement.
He added that with the Greek vote outcome, Europe was headed for unfamiliar scenarios “which will require from all a sense of responsibility and a long-term and strategic vision”.
Meanwhile, Renzi has insisted that Italy was not likely to be affected by fallout from any eventual Greek implosion.
“We are not saying that everything is going to be fine, we only say that the work that has been done in the last few months puts Italy in a (better) condition than it was in the past. We are no longer in the dock, we are no longer spoken of in the same breath as our unfortunate Greek friends.”
In spite of Renzi's calm, the referendum has become something of a political football in Italy. Beppe Grillo, leader of the populist M5S party even went to Athens this weekend to show his support for the Greek people.
While there, he published a video on his blog in which he conversed with a statue of Thermistocles, one of the founders of democracy. “I really hope 'no' wins,” said Grillo in the video. He also criticized the amount of “disinformation” he believed there had been in the media in the lead up to the vote.
Matteo Salvini, leader of the far-right Lega Nord party, was quick to have his two cents and took to Twitter this morning to criticise the Italian Prime Minister's response to the 'no' vote. “From Matteo Renzi not even a word,” read the tweet, “Why do you think the 'phenomenon' has stopped tweeting?”
— Matteo Salvini (@matteosalvinimi) 6 Luglio 2015