Some in the crowd ripped away bike racks in the area of Pennsylvania Avenue Plaza.
"An individual inside the vehicle began firing gunshots at contract security officers for the Federal Protective Service of the Department of Homeland Security," FBI San Francisco said in a statement.
The man had been caught by the trailer tire, pulled under and was eventually run over by the tire, authorities said.
The president's tweeted remarks are a stark contrast to how he has called on other world leaders to respond during times of civil unrest.
"We have yet another innocent man being killed by a police officer," said Anzhane Laine in Washington, D.C.