The ambulance service was called to the incident in King Street, Dudley, at 1605 BST on Saturday.
West Midlands Police said initial witness accounts had suggested that a family car had become caught up in a small pocket of protesters.
No-one received life-threatening injuries in the incident.
A force spokesman said the investigation was in its very early stages.
He added: "Whilst trying to work its way clear the car collided with some of the protesters as they left the protest area."
The force said on Sunday it was starting a separate post-event investigation into disorder and criminal offences which happened in Dudley on Saturday.
Twenty-one arrests were made during a police operation - 17 for violent disorder, two for affray, one for a public order offence and one for possession of an offensive weapon.
The force said it was starting inquiries into incidents reported, including criminal damage to buildings and incidents of disorder.
Anti-fascist group Unite Against Fascism held a counter-protest against the English Defence League, which says it is against Islamic extremism in the UK.
BBC News
