The announcement on Sunday came after a meeting between Conte and Italian President Sergio Mattarella to discuss a proposed list of cabinet members, in what would have been Italy's first populist government.
In a televised address, Mattarella said he had rejected the candidate, 81-year-old eurosceptic economist Paolo Savona, because he had threatened to pull Italy from the single currency.
University of Florence law Professor Giuseppe Conte received a mandate last week from staunchly pro-Europe President Sergio Mattarella to try to form viable government out of rival populist forces.
Conte said he tried his hardest to form a government and had full cooperation from would-be coalition partners, the populist Five Star Movement (M5S) and League parties.
"Giuseppe Conte has given up the mandate to form a government, given to him on May 23", an official from the presidential palace said.More news: Fulham back in Premier League with playoff win over Aston Villa
Di Maio's Movement is the largest party in the new Parliament but remains far short of having an absolute majority.
The chances of the economist gaining approval for any technocrat government are slim, as Five Star and the League boil with anger at their own coalition stumbling on the home straight.
Breitbart News former executive chairman Stephen K. Bannon, who is now in Rome, had been invited by the Center for American Studies to give a public address on Italy's historic populist moment Monday afternoon, but organizers cancelled the event Sunday night out of fear of riots.
A former judge of Italy's constitutional court, Mr Mattarella has refused to bow to what he saw as "diktats" from the two parties which he considered contrary to the country's interests.
Polls have suggested that the League, which won 17 percent of the vote in March, would surge in any early ballot, while support for 5-Star remained strong on around 35 percent.More news: ‘Solo’ Lacks `Star Wars' Mojo as Film Shows Lackluster Sales
Mr Mattarella said he was considering political party leaders' requests for another election and would announce his next move "in the next hours".
League leader Matteo Salvini tweeted Sunday that he would keep fighting "to the end" for the anti-euro candidate he wants to be economy minister. Under that clause, parliament can seek to remove a president if a simple majority of lawmakers votes in favour.
"I don't want to talk about impeachment", Salvini said speaking in a radio interview with Radio Capital.
Mr Di Maio also scathingly criticised Mr Mattarella's veto of Mr Savona as "incomprehensible".
Savona has had high-level experience at the Bank of Italy, in government as industry minister in 1993-94, and with employers' lobby Confindustria.More news: Meghan Markle reportedly made her wedding guests cry with a love poem
Mattarella's claims that Savona is a danger to Italy's stay in the eurozone lack any ground, Stefano argued, since Savona's current stance is that Italy can assert its place in European economics without necessarily abandoning the euro.