Kazakhstan's 2022 Wimbledon champion was scheduled to play Ponchet, who is making her U.S. Open main draw debut this year, in the fourth and final match on Court 17.
"I did not want to finish the last Grand Slam of the year this way but I have to listen to my body, and I hope I can close out the remainder of the year strong," said Rybakina.
"Thank you to all the fans who have and continue to support me, and I wish the remaining players good luck."
Since reaching the Wimbledon semi-finals last month, Rybakina withdrew from the singles and mixed doubles events at the Paris Olympics, lost her opening match in a U.S. Open tune-up event and announced a split from her long-time coach.
Ponchet, who is ranked 143 in the world, will now face either former Australian Open winner Caroline Wozniacki or Mexico's Renata Zarazua in the third round.
