"Following the results of the tests carried out today, we have decided that the wisest thing to do is to be cautious and not participate in Rome or Roland Garros," Alcaraz wrote on Instagram.
"It's a difficult moment for me, but I'm sure we'll come back stronger."
The 22-year-old star became the youngest male player to complete a career Grand Slam when he won the Australian Open in January, giving him titles at all four Grand Slam tournaments.
The Spaniard lost the top spot in the world rankings to Jannik Sinner of Italy earlier this month.
The French Championships start on May 18.
