"They've replaced a government with a majority with one that won't obtain one", said Di Maio.
"Sticking with the euro is a fundamentally important choice for our country and our young people", Mr Mattarella said in a late-night statement at the Quirinale Palace.
Cottarelli first joined the IMF in 1988, following six years in the Bank of Italy's Monetary and Financial Sector Division.
The country, Europe's fourth biggest economy, has been without a government since the General Election in March which saw populist left and right movements League and 5-Star win the most votes.
After the coalition's collapse, Italian bonds, stocks and the euro rallied.
The spread between Italian and German 10-year yields - a measure of the premium investors demand which is highly sensitive to political risks - fell back below 200 basis points, after widening dramatically in the past month to hit heights last seen in April last year. It trimmed some gains to stand 0.4 percent up on the day at $1.1696.
Five Star and League parties have said they will support earlier elections, and electoral polls indicate they would garner even more votes a second time around.
In televised remarks to the nation, Mattarella explained that he could not approve of a finance minister who would "probably or nearly certainly bring about Italy's exit from the eurozone".
"The main risk is that the stand-off will further embolden the 5-Star Movement, and, especially, the League". Such a call is usually a prelude to being offered a mandate to form a government.
Five Star had been trying to form a government with the right-wing League.
Former left-wing prime minister Massimo D'Alema already warned on Saturday that "if we have to go back to the ballot box because of a veto of Savona, Salvini will pick up 80 percent" of the vote.
Mr Mattarella's office declined to reveal his plans.
"We will ask parliament to charge Mattarella with high-treason because he has acted under foreign pressure", Brothers of Italy chief Giorgia Meloni said on La7 television channel.
In an interview for the TV program "Che Tempo Che Fa", Five Star leader Luigi Di Maio said he was "asking for Mattarella's impeachment".
She continued: "I am confident that the Italian institutions and the president of the republic will prove to be as always serving the interests of the Italian citizens that by the way coincides also with the strength of the European Union". The constitutional court would then be called to decide whether to enforce the decision. "If you want to talk about it, we need to do it openly and with a serious, in-depth analysis".