Police were questioning the driver to try to determine whether he deliberately attempted to smash the barrier down or whether he simply lost control of the car. He was being held on suspicion of causing criminal damage.
The tourist, a woman, was taken to a nearby hospital with leg injuries. The gates were damaged, but the car did not break through, police said.
The prime minister and his family were not at home at the time of the incident, police said.
The gates block access to Downing Street, a short street at the end of which sits No. 10, the prime minister's office and home.
Peter Saacs, 55, who was driving a bus past Downing Street, said he saw the crash.
"The driver (of the car) pulled up as if to show someone Downing Street," he said. "Then he was wheel-spinning in front of me and pulled out and went round the back of our bus and into the gates."
A support on the heavy iron gates was broken in the crash and the bolt connecting the gates to the ground was bent.
Neither the driver of the car nor the tourist was immediately identified.