The Federal Aviation Administration had granted him permission for his first U.S. flight, but Yves Rossy scrubbed the mission less than an hour before he was scheduled to take off, saying he wants to do test runs before attempting the flight.
The 51-year old had planned to jump from a helicopter near eagle point, a rock formation that resembles a bird.
He said he would still like to do the stunt, but he has not yet rescheduled it.
Rossy has soared over the Swiss Alps and the English Channel in his jet-propelled suit.
In announcing his decision to scrap the flight, Rossy pointed to the canyon's rugged walls, swirling air currents and cactus-covered cliff sides.
"It's the most challenging place I could fly," he said. "I don't know any professional aviator who will give a show without training."
The Associated Press contributed to this story.