In its latest update, the government of the Liguria region tweeted that 19 bodies had already been identified and 15 people had been admitted to hospital, nine in critical condition.
Some 200 firefighters were on the scene, the fire service said, and Italian news agency ANSA said two survivors had been pulled out of the rubble under the collapse.
Notwithstanding the importance of a road that sees 25 million vehicles pass along it every year, the demolition of the bridge was being studied as far back as 2009.
Officials evacuated several hundred people living along the raised highway that traverses the city as a precaution.
On Tuesday specialist engineering website Ingegneri.info published a piece that highlighted how the bridge had always presented "structural doubts", calling it "a tragedy waiting to happen".
Firefighters heard voices under the rubble and pulled out seven survivors, fire official Bruno Frattasi told the state broadcaster RAI. "We hope to find more people alive". He said they were still trying to figure out the cause of the collapse. "Obviously for me it is not", said Deputy Prime Minister Matteo Salvini, who leads the right-wing League which governs with the 5-Star Movement.
The government has already pledged to increase public investment and lobby the European Commission to have the extra spending excluded from EU deficit calculations.
"Italians have the right to modern and efficient infrastructure that accompanies them safely through their everyday lives", Mattarella said in a statement.
Photographs showed traffic still travelling along the A10 toll highway, with drivers unaware of the disaster ahead - and one lorry perched yards from the edge of the destroyed bridge.
"Suddenly the bridge came down with everything it was carrying". The highway operator said work to shore up the foundation of the bridge was being carried out at the time of the collapse, adding that the bridge was constantly monitored.
Civil Protection authorities said at least 30 cars and three heavy vehicles were on the 80-metre section of the span that collapsed in the industrial area of warehouses.
"It's not acceptable that such an important bridge. was not built to avoid this kind of collapse", Deputy Transport Minister Edoardo Rixi said on SkyNews24, speaking from Genoa. It called for a "Marshall Plan" to fix or replace tens of thousands of bridges and viaducts built in the 1950s and 1960s.
Transportation Minister Danilo Toninelli called the collapse "an enormous tragedy", adding that if negligence played a role, "whoever made a mistake must pay".
Autostrade, a unit of the infrastructure group Atlantia, controlled by the Benetton family, manages the section of the toll highway that collapsed.
Stefano Marigliani, the Autostrade official responsible for the Genoa area, told Reuters: "The collapse was unexpected and unpredictable".
He said there was "no reason to consider the bridge was unsafe".
He said there had been no indication that the bridge was risky.
"There has not been sufficient maintenance and checks, and safety work for many bridges and viaducts and bridges in Italy constructed - nearly all - during the 1960s", he said. "But there will be a time when the cost of maintenance will be higher than a replacement", he told Italian media Primocanale.
Italian Premier Giuseppe Conte visited the city and called the collapse "a serious wound for Genoa, Liguria and Italy".