Injured Draper takes time out as he targets US Open
Wimbledon champion Jannik Sinner and 24-time Grand Slam champion Novak Djokovic have also pulled out of the Canadian Open, which starts on July 27.
The Cincinatti Open begins on August 7 and wraps up on August 18, six days before the start of the US Open, the final Grand Slam of the year.
Both tournaments are Masters 1000 events, the level below the four majors.
World number five Draper, who was knocked out in the second round at Wimbledon by Marin Cilic, said his injury was not serious and he would be fit for the US Open in New York.
"After Wimbledon I picked up an injury in my left arm, nothing serious, but I have to make sure it recovers fully for the rest of the season," he said on X.
"Unfortunately, I won't be able to compete in Toronto and Cincinnati... See you in NYC!"