He spoke briefly with the three occupants and then hit their car again as he departed, investigators said. They followed him while calling police.
Bryson was cited for felony hit-and-run, although he has not been charged.
Bryson then caused a second collision in the nearby city of Rosemead, striking a car occupied by a man and a woman, authorities said. Bryson was found alone and unconscious in his car.
White House chief of staff Jack Lew spoke to Bryson on Monday, press secretary Jay Carney said. Carney said the administration was in the process of gathering information about the incidents.
"We're obviously concerned about the incident, about the health-related issues that played a role in this incident," Carney said.
Authorities did not know whether Bryson had a prior medical condition.
Bryson voluntarily took a Breathalyzer test after the crashes that did not detect any alcohol, but investigators also were awaiting the results from a blood test, said Los Angeles County sheriff's Capt. Mike Parker.
Bryson was treated at the scene before being taken to a hospital. Two people in the first collision were treated by paramedics after complaining of pain, authorities said. The couple involved in the second crash also complained of pain but declined medical aid.
Jerome Engel Jr., a neurologist at the University of California, Los Angeles, said such an episode could be consistent with someone who has suffered a series of epileptic seizures.
After a seizure, a person is often confused, and that state of confusion can last for a while. "You may even seem to be alert and awake, but you're not really behaving normally," Engel said.