The world number two sustained the injury to his right wrist during his opening-round victory over Otto Virtanen in Barcelona, before pulling out of the tournament straight after.
Following some scans, he has now decided to miss the second major of the year in Paris, as well as the upcoming Masters 1000 in Rome - a tournament he also won in 2025.
"After the results of the tests carried out today, we have decided that the most prudent thing is to be cautious and not participate in Rome and Roland Garros, while we wait to assess the evolution to decide when we will return to the court," Alcaraz said on social media.
"It's a complicated moment for me, but I'm sure we'll come out stronger from here".
Alcaraz defeated Jannik Sinner in an all-time great match in last year's final, which lasted five and a half hours, and overcame Alexander Zverev in five sets the year before.
However, he will not get the chance to defend his crown at Roland Garros, making Sinner the heavy favourite to win his maiden French Open title and complete the Career Grand Slam.
Alcaraz achieved that feat himself in January at the Australian Open, becoming the youngest man in the Open Era to clinch all four Slams.
Sinner, who reclaimed the world number one ranking from Alcaraz after beating him in the Monte Carlo final just under two weeks ago, can also extend his lead at the summit with the Spaniard out of action.