The Italian said he was "sad and disappointed" to drop out, adding that he had started feeling unwell on Monday after a practice week in Monaco.
Nadal, the 2008 singles gold medallist, will pair up with Alcaraz in a tennis dream team at Roland Garros, looking to also add to the doubles title he won with close friend Marc Lopez at Rio in 2016.
The 37-year-old, a two-time Olympic champion, is planning to play in the singles and doubles at his fifth Games.
The 38-year-old Spanish left-hander has a protected ranking of ninth to put himself into the field for the Flushing Meadows fortnight, which begins on August 26.
"I don't have any limitations in my mind. I still want to keep going and play as long as I feel like I can play on this high level."
"So let's see if at the end of my career it's going to be 25, 30, 15, four (Slams). I don't know. All I want to say is I want to keep enjoying and let's see what the future brings to me."
Spain's Carlos Alcaraz produced an utterly ruthless performance to dismantle Serbia's Novak Djokovic 6-2, 6-2, 7-6 (7/4) in a rematch of last year's Wimbledon final to successfully defend his title on Sunday.
Djokovic will be playing in his 10th Wimbledon final and 37th at the 75 majors in which he has played.
Barbora Krejcikova of the Czech Republic defeated Italy's Jasmine Paolini to win the Wimbledon women's title on Saturday for her second Grand Slam singles triumph.
Daniil Medvedev bizarrely called umpire Eva Asderaki a "small cat" during the Wimbledon rant that earned the Russian a code violation in his semi-final loss against Carlos Alcaraz on Friday.
Italy's Jasmine Paolini has set her sights on capping a "crazy" run to Saturday's Wimbledon final against Barbora Krejcikova by winning a maiden Grand Slam title. Meanwhile, Krejcikova will be hoping to win a second Grand Slam final of her career after winning the French Open in 2021.
Carlos Alcaraz wants to play his part in a "perfect" sporting Sunday for Spain as he targets Wimbledon final victory against Novak Djokovic before his country face England in the Euro 2024 title match.
Novak Djokovic has backed Carlos Alcaraz to rack up more Grand Slam titles, but the Serb hopes that run doesn't start when they clash in a blockbuster Wimbledon final on Sunday.
With Centre Court still filling up following Alcaraz's 6-7(1), 6-3, 6-4, 6-4 victory over fifth seed Daniil Medvedev a short while earlier, second seed Djokovic broke after a breathtaking 26-shot rally in the sixth game of the opening set.
World number three Alcaraz beat his fifth-ranked opponent 6-7 (1/7), 6-3, 6-4, 6-4 and will face either seven-time champion Novak Djokovic or Lorenzo Musetti for the title on Sunday.
Victory on Friday would take Djokovic to a 10th final at the tournament and 37th at the Slams.
Krejcikova dug deep for victory against fourth seeded Rybakina in two hours and seven minutes on Centre Court.
Italy's Jasmine Paolini reached her first Wimbledon final with a stirring fightback to beat tearful Croatian Donna Vekic 2-6, 6-4, 7-6 (10/8) on Thursday.