As a result, Giuseppe Conte, a law professor who had been named as the coalition's prime minister, resigned after a day of tense talks with Mattarella at the Quirinal Palace in Rome.
"Giuseppe Conte has given up the mandate to form a government, given to him on May 23," an official from the presidential palace said.
In a terse comment to reporters, Conte said he "gave the maximum effort, attention, to carry out this task with the full collaboration" of the Five Star Movement and League.
Luigi Di Maio, the youthful head of Five Star, demanded the impeachment of Mattarella over his veto.
He called into an Italian television programme on Rai, the state broadcaster, to say he should be put on trial for sparking an "institutional crisis".
He also took to Facebook to tell his supporters that he and Matteo Salvini, the leader of The League, had been ready to form a government tomorrow.
He even read out the full list of ministerial appointments that the two leaders had put forward.
"I am very angry. You can imagine how much time we spent trying to form a government over the last 80 days," he said.
"We worked day and night to give a government to this country," Di Maio said.
He said the coalition had been torpedoed before it got off the ground by "the credit rating agencies" and "the financial lobbies" – a reference to the concern in Brussels and elsewhere over the appointment of Savona as economy minister.
He said the President's veto was "unacceptable", describing it as "an institutional clash without precedent."
"What's the point of going to vote if it's the ratings agencies that decide?" he asked.
Salvini of the League was also furious about the presidential veto.
"We will not be blackmailed by anyone," he said in an angry speech from the city of Terni in Umbria.
"If a certain minister bothered Berlin, or a certain minister bothers the powerful forces that have massacred us up until now, that means it was the right minister."
Savona, 81, has described the common currency as a "German cage" for Italy.
He has accused Berlin of trying to achieve by economic means what it failed to do with armies of stormtroopers in World War II – the domination of Europe.
He has claimed that Germany has "the same objective today as did Funk," a reference to Walther Funk, who served under Hitler as Reich minister for economic affairs and was convicted as a war criminal at Nuremberg.
He has also called for a "plan B" to be drawn up to allow Italy to exit the euro-zone with as little damage as possible, should that should prove necessary.
Mattarella said he refused to approve Savona because the appointment would have "alarmed markets and investors, Italians and foreigners."
Speaking to the media after the dramatic collapse of the nascent coalition, he said: "Every day, the (bond) spread goes up, it raises our debt" costs.
The ministerial appointment would "have probably or more likely inevitably led to the exit of Italy from the euro."
In a televised address, a sombre Mattarella said he was not to blame for the impasse, adding that he had proposed alternatives for the key position, but his suggestions had been spurned by Five Star and the League. "It is my duty to protect the savings of Italians," he said.
With the government in tatters before it was even sworn in, Mattarella could now appoint a caretaker government until new elections can be held.
