The incident occurred in the busy Fisherman's Wharf area of San Francisco when the cyclist rode up to the vehicle, yelled at the driver, banged on his window and pushed in his mirror, police spokesman Carlos Manfredi told AFP.
"During the exchange the cyclist sped up and the vehicle also sped up, striking the back of the bike," Manfredi said.
"The cyclist sustained a punctured lung, broken ribs and a broken clavicle," he added.
The driver, Emerson Decarvalho, 38, was charged with assault with a deadly weapon after the incident, which took place Sunday afternoon.
He was also charged with failing to maintain a three-foot (one meter) space between his car and the cyclist.
Uber offered to help in the investigation of the incident.
"Our thoughts are with the victim of this terrible incident. We are gathering more information, have reached out to the San Francisco Police Department and will assist their investigation however we can," said Uber spokeswoman Kate Downen. Uber said there was no passenger in the vehicle at the time.
The San Francisco Chronicle newspaper said the cyclist was knocked unconscious and taken to hospital.
Police said it was not clear what provoked the confrontation and said Decarvalho remained in custody.
Manfredi said the altercation took place in one of San Francisco's busiest neighborhoods.
"The location where it occurred is a highly dense area where people go out and a lot of tourists were out and about," he said.
Uber drivers must be 18 years old to join the service must agree to the company's terms and conditions, including agreeing to "comply with all applicable laws when using the services."
The San Francisco company also requires that drivers do not "cause nuisance, annoyance, inconvenience, or property damage."
About one million Uber trips are carried out daily in 295 cities in 55 countries.
The company announced a worldwide code of conduct this year after an Uber driver was charged with rape in India in December.
An Uber driver was arrested in Denver this month after he tried to rob the home of a customer he had dropped off at the airport.
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