The world number three from Spain beat Tallon Griekspoor of the Netherlands 6-1, 7-6 (7/3) to stay on course for a gold medal to add to his four Grand Slam titles.
The 21-year-old Alcaraz is back on court on Tuesday alongside Nadal when he will again face Griekspoor, who partners Wesley Koolhof.
"It's a pain that I've been dealing with the last tournaments so I know what I have to do," Alcaraz said of his groin niggle.
"It's normal for a tennis player as this game is so demanding.
"I will try to recover and be ready and at 100 percent in the doubles."
Alcaraz, the reigning French Open and Wimbledon champion, needed to leave the court to have treatment on his right leg late in the second set and then had to save a set point.
In the next round of singles he will face Roman Safiullin, the Russian player competing at the Olympics as a neutral.
Earlier Monday, Alcaraz had seen Nadal swept off court by old rival Novak Djokovic in straight sets in the pair's 60th meeting.
The defeat ended injury-plagued Nadal's outside hopes of winning a second singles gold after his 2008 triumph in Beijing.
But Alcaraz can still help his compatriot win a second doubles gold after Nadal teamed with Marc Lopez at the 2016 Rio Games.
"When Djokovic is playing at this level it's really, really difficult to play against him," said Alcaraz.
"He was really aggressive, made no mistakes, moving very, very well. I saw Rafa playing let's say not his best tennis for sure.
"But when Djokovic is like this, he was the better one."
Griekspoor, 28, lost for the third time in three meetings with Alcaraz and has yet to win a set off the Spaniard.
"I didn't see any problems with him," said the 28th-ranked Dutchman.
"He went for every shot in the tiebreak. It's not like he fell down or twisted his ankle and that he couldn't walk anymore. For me, he was still all over the place."
