"I know the way that I have to deal with this pain, it is normal for a tennis player, the schedule is so tight, so demanding, so I will try to recover as soon and as well as I can tonight to be ready at 100% tomorrow in the doubles," the Spaniard told reporters.
Alcaraz is set to play with Rafa Nadal in the second round of the Olympic doubles tournament.
"It is in the adductors, so it is a muscle that requires a lot in the movement, but I will see tonight," he said.
Despite suffering from some discomfort, the manner of his victory proved that Alcaraz will be hard to beat at Roland Garros.
The world number three, who won this year's French Open and Wimbledon titles, was back on Court Philippe Chatrier and set the tone from the start with an ace and two forehand winners.
With the crowd having seen Nadal beaten by Novak Djokovic, Alcaraz appeared determined to show he was ready to take up the baton, breaking Griekspoor immediately and racing through the first set in 32 minutes.
But the Dutchman, ranked 28th in the world, made a better fist of it in the second set. He repeatedly held his serve under intense pressure, giving back as much as Alcaraz threw at him and at times forcing the Spaniard into some uncharacteristic errors.
When the match went to a tiebreak, Alcaraz stepped up the pace and showed his subtlety to win the match.
"He made it very difficult for me in the first set," Griekspoor told reporters.
"I think I did pretty well in the second, made a good fight of it, made a battle of it and got a small chance at the end of the second, but he did well so all credit to him for playing a great match."
In the third round, Alcaraz will play Russian Roman Safiullin, the world number 66, who is competing as a neutral athlete.
The set went to a tiebreak with Alcaraz stepping up the pace and showing subtlety to win the match.