The 23-year-old Spaniard had been slated to make his return in the key tune-up for the US Open, the final Grand Slam of the year that starts on August 30 in New York where Alcaraz is the defending champion.
Organizers said in a statement that Alcaraz had withdrawn from Cincinnati with "an ongoing wrist injury."
"We know Carlos is doing everything he can to get back to playing tournaments as soon as possible," tournament director Bob Moran said. "We wish him the best with his recovery and look forward to welcoming him back to Cincinnati in the future."
Alcaraz had planned to take the court in Cincinnati for the first time since he was injured in the first round of the Barcelona Open in April.
The seven-time Grand Slam champion pulled out of tournaments in Madrid and Rome and then Roland Garros, where he was the reigning two-time champion.
He also missed Wimbledon, where his rival Jannik Sinner prevailed for the second year in a row against Germany's French Open winner Alexander Zverev.