President Sergio Mattarella was speaking at the Quirinal Palace in Rome, ahead of elections next month which are forecasted to give no party a majority.
Mr Mattarella warned “without a sense of community, violence is around the corner.”
Violent clashes have broken out in Italy in recent days with the shooting in Macerata and protests on the streets of Rome after Turkish President Recep Erdogan visited the Vatican.
Alleged gunman Luca Traini is being held on suspicion of attempted murder inspired by racial hatred, after five men and one woman were allegedly targeted in drive-by shootings.
Italian police have released footage of Traini’s bedroom that shows Hitler’s autobiography Mein Kampf and other publications linked to Nazism.
Witnesses said Traini, 28, made a fascist salute as he was arrested with an Italian flag draped over his shoulders.
The six injured are aged between 22 and 33 and are originally from Ghana, Gambia, Nigeria and Mali, according to the Italian broadcaster RAI.
Police commander Michele Roberti told Sky TG24 the shooter was “lucid and determined, aware of what he had done.”
He said: “It’s likely that he carried out this crazy gesture as a sort of retaliation, a sort of vendetta.”
The drive-by shootings took place just days after a Nigerian man was arrested over the death of an 18-year-old Italian woman, whose mutilated body was found packed into two suitcases near Macerata.
Italy has reportedly received more than 600,000 migrants by boat over the past four years.
Matteo Salvini, head of Italy’s anti-immigration Northern League, has vowed to deport 150,000 migrants in his first year if he is elected.
And Italy’s disgraced former Prime Minister Silvio Berlusconi has pledged to deport 600,000 illegal immigrants if his Forza Italia party is elected.
EU Commission Vice President Frans Timmermans posted on Twitter after in Macerata: “A wilful attack on our most fundamental values, an attempt to destroy the fabric that binds us together as Europeans.”
“It is our duty to condemn this violence and the despicable ideology that underpins it.”
(Additional reporting by Maria Ortega.)