"His condition is life threatening," Goossens said.
If he survives, he will be charged with attempting an attack on the royal family, and murder.
He had no record of mental illness or previous crimes.
"There are no indications of terrorist links," Goossens said.
Several members of the royal family looked on in horror as the car drove through the crowd, which had gathered in the city to celebrate the Queen's Day national holiday.
"Her majesty the queen, on behalf of other members of the royal family, expresses her shock at the incident and conveys her condolences to the victims and their next of kin," a statement on behalf of the queen said.
The Dutch ministry of royal affairs later said celebrations had been cut short by the incident.
It was not clear how the car managed to enter the parade area, which had been sealed off by police hours earlier.
Hundreds of thousands of people flocked to cities across the Netherlands on Wednesday night and Thursday to celebrate the national holiday, which was originally intended to celebrate the birthday of Queen Juliana, Beatrix's mother.