The royal, his wife and her mother were in a car chase involving paparazzi that is said to have lasted for more than two hours.
Harry and Meghan attended an awards ceremony in New York on Tuesday, and the incident occurred after that.
A spokesperson said: “Last night the Duke and Duchess of Sussex and Ms Ragland were involved in a near catastrophic car chase at the hands of a very aggressive paparazzi ring.
“This relentless pursuit, which lasted for more than two hours, resulted in multiple near collisions involving other drivers on the road, pedestrians, and two NYPD officers.
“While being a public figure carries a level of public interest, it should never come at the cost of anyone’s safety.
“The dissemination of these images, given the way in which they were obtained, encourages a highly intrusive practice that is dangerous for all involved.”
The Duke and Duchess of Sussex were joined at the award ceremony by Meghan’s mother, Doria Ragland, at the Ms Foundation for Women’s 50th Anniversary Gala event.
The Duchess received the Ms Foundation’s Women of Vision Award, in recognition of her “global advocacy to empower and advocate on behalf of women and girls.”