Police said seven people were hurt in the incident, which took place as residents and visiting soccer fans thronged the centre of Moscow on a balmy summer evening.
The witness added some of those hit were wearing Mexican team colours.
He also is suspected of having had too much to drink, according to the Russian news agency Interfax.
The authority cited the driver as saying he had not driven into the crowd on objective.
Mexico are due to play Germany at the Luzhniki Stadium on Sunday in the city and two Mexican women are in a stable condition after the incident, the country's embassy in Moscow said on Twitter.More news: Major fire ravages celebrated Glasgow School of Art for second time
There is now no details on the condition of those who were injured.
Police have detained a taxi driver who had a Kyrgyz driving license, a Moscow traffic authority wrote on social media.
The authority cited the driver, who was in police custody, as saying he had not driven into the crowd on objective.
Policemen stand guard after a taxi ran into crowds of people in central Moscow, Russia.