Alcaraz sustained the injury during the final of the Barcelona Open where he was denied a third title by Denmark's Holger Rune, who stunned the Monte Carlo Masters champion in straight sets.
"I couldn't practice the entire week and found out I had hamstrings and groin injuries. I needed to listen to my body and make the right decision," Alcaraz told reporters.
"On Monday I'll have more medical tests and after that we will have a better idea on how long I'll be out.
"This tournament is very exciting, but things haven't worked out the way I wanted to be able to compete. It hurts me not to be able to play here in Madrid in front of my people, my friends and my family."
Alcaraz had received a bye into the second round and was scheduled to play his first match on Friday.
He took a medical timeout during the Barcelona Open final and continued to receive treatment in the next changeover before Rune prevailed, dropping the Spaniard to number three in the world rankings.
The next major tournament in the claycourt swing is the Italian Open, a Masters 1000 event from May 7th to 18th in Rome, where world number one Jannik Sinner is expected to make a comeback from his doping ban.
The French Open main draw gets underway on May 25th.
"I expect to go back to court and start practising in a couple of weeks. Rome is still a possibility, but for sure I'll be in Paris," Alcaraz added.
"It's not an easy decision because Madrid is the first tournament I have ever been to as a young kid, it's the most special place for me to play.
"But (prioritising Roland Garros) was a big part of my decision to stop and recover properly."