Protesters 'beaten up'
The bridge, which crosses the Rhine River, is 5km from the conference centre where 28 Nato leaders, including Barack Obama, the US president, were meeting.
Black smoke could be seen clearly from the venue.
About 50 of the protesters were reported to be hurt.
Organisers said the majority of demonstrators in Strasbourg and nearby Baden-Baden did not aim to carry out violence and blamed tensions on security forces.
"No one here has attacked any police, but we have been hit with teargas and beaten up," Monty Schaedel, a protest spokesman, said.
Demonstrators from as far as Japan flocked to Strasbourg to campaign against war, capitalism, defence spending and nuclear weapons in view of the assembled leaders.
The initially calm protest on Saturday had turned violent by noon, when protestors first clashed with police at the Europe Bridge.
Some rioters later set fire to a former customs station on the French side and an Ibis hotel, which were both gutted before fire services were able to put them out.
On the German side, around 5,000 demonstrators gathered peacefully to cross the Europe Bridge, before finding themselves blocked by scores of police backed by water cannons.
The Czech Republic has also boosted security, ahead of a visit by Barack Obama, the US president, who headed to Prague on Saturday following the Nato summit and will meet with EU leaders there over the weekend.
France and Germany, which co-hosted the Nato summit, each deployed 15,000 police and troops at the start of the two-day meeting, in a security operation costing about $150m.
Police in Strasbourg had arrested hundreds of protesters before the summit opened on Friday.
Schools and the university were closed and most businesses and restaurants have shut down for the duration of the summit.