The fashion designer, 45, has since apologised after she left the scene without handing over contact information.
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But when he started filming her as she got back into her car, she appeared to panic and told him to take her licence plate number before driving off.
"Just do whatever you need to do. Take the number of my licence plate. Do whatever you need to do, no problem," she told him - assuming he could file a claim from her licence plate - in video footage obtained by The Sun.
He then followed the designer to a nearby school to try and get her details and there she reportedly accused him of harassing her.
When Arash pointed out that she had refused to give her details to him, Stella replied: "I know, you were filming! You're filming now! Would you stop filming!"
The designer eventually took his number and the following day he received a call from her husband who passed on details of their insurance broker and the necessary details to make a claim.
Arash, who said he hasn't been able to drive since the incident and has lost out on earnings, told the newspaper: "She's very wealthy. She shouldn't have done this to me. It's very unfair."
Stella's spokesman said: "Ms McCartney has never been in this type of incident before. Although she and the taxi driver did not swap full details, Ms McCartney thought an insurance claim could be made from the licence plate number she was happy to provide and knew the driver had a note of.
"She thought she had done the right thing. When the driver, with two passengers, started filming her she felt intimidated - particularly when she realised they had followed her to the kids' school, still filming her.
"She accepts liability for the incident and is very apologetic for any issues that have arisen."