Footage on social media shows her legs trapped between the vehicles at a busy junction in Grosvenor Place, near Hyde Park Corner, in central London.
The Metropolitan Police said officers are "investigating an allegation that a Land Rover was deliberately driven at the woman, trapping her against a Mercedes car".
The force said officers were called to the scene at 8.15pm on Sunday and that a 23-year-old man - the suspected driver of the Land Rover - was arrested on suspicion of affray.
The force added that a Taser was used to detain the man, who did not require hospital treatment - and that he remains in custody.
He was later further arrested on suspicion of grievous bodily harm and driving whilst disqualified.
Deepak Anand, 40, a tourist visiting from Vancouver, Canada, said he was on a bus with his wife and son when he saw what happened on Sunday evening.
He said he spotted a man trying to pull a driver out of a Land Rover, which then "accelerated on to oncoming traffic" before lodging the woman between the car and a black Mercedes-Benz.
The Mercedes also collided with the bus, but there were no reports of any injured passengers.
Onlookers said the woman could be heard screaming and fell to the ground after the cars were separated.
Mr Anand said his son had been shaken by the incident, adding: "I'm not sure that we'd choose to visit London as a matter of priority in the future given the safety concerns, particularly with our son."
The woman, in her mid-40s, was taken to a central London hospital for treatment.
Her condition is not life-threatening, but police say it may be life-changing.
A man who complained of feeling unwell at the scene was taken to hospital, but has since been discharged.
:: Any witnesses, anyone with dashcam footage, or anyone with any information is asked to call police on 101 or contact the force via Twitter @MetCC quoting CAD 6953/11Aug.