Richie complained of pain but opted to seek her own treatment and no ambulance was called, Hill said.
Edwardo Arrivebene, 29, a freelance photographer, was arrested on suspicion of driving without a license, Hill said.
Hill did not know if there were any passengers in either vehicle.
Richie gave birth last month to Sparrow James Midnight Madden, her second child with Joel Madden. Their daughter, Harlow, will be 2 in January.
Richie became a clothing and accessories designer after starring with Paris Hilton on "The Simple Life." Madden, 30, fronts the band Good Charlotte and runs the DCMA Collective clothing line with his twin brother, Benji.