Her withdrawal comes ahead of her second-round match against Number 16 Beatriz Haddad Maia.
The unseeded Osaka is the defending champion having won in her birthplace Osaka in 2019 before COVID-19 shut down the tournament for two years.
Four-time Grand Slam champion Osaka, a winner at Flushing Meadows in 2018 and 2020, was beaten 7-6 (7/5), 6-3 by 19th-seeded American Collins in a battle of two big hitters.
Novak Djokovic is missing a Grand Slam tournament for the second time this season because he is not vaccinated against COVID-19.
Osaka was trailing 7-6 (4), 3-0 against Estonia’s Kaia Kanepi when she withdrew from the contest.
Coco Gauff held off four-time major champion Naomi Osaka 6-4, 6-4 on Thursday night to advance to the quarterfinals at the Mubadala Silicon Valley Classic.
Osaka hit 11 aces and saved 7 of 8 break points in the hard-court tournament that serves as a tuneup for the U.S. Open.
The 23-year-old Rybakina won her first major title at Wimbledon on Saturday.
The four-time major champion and former No.1-ranked player posted a photo of herself on a grass court and wrote on Twitter: "my Achilles still isn't right so I'll see you next time."
As rain fell on the outside courts, Swiatek required just 54 minutes to dispatch Ukrainian qualifier Lesia Tsurenko 6-2, 6-0 under the roof on Court Philippe Chatrier.
The former world number one pulled out of last year's French Open ahead of the second round, citing mental health issues.
Osaka kicked off her clay campaign with a comfortable 6-3, 6-1 win over qualifier Anastasia Potapova in the first round on the Madrid Open center court on Friday.
Swiatek will replace the now-retired Ashleigh Barty at number one in the women's rankings when the points are updated on Monday.
The unseeded Osaka defeated No. 22 Belinda Bencic 4-6, 6-3, 6-4 on Thursday in the Miami Open semifinals.
The Russian held the No. 1 ranking briefly, lost it to Djokovic again about a week ago and is now closing in on capturing it again.
Shaking off a slow start, the top-seeded Medvedev reached the quarterfinals of the Miami Open on Tuesday by defeating Jenson Brooksby of the U.S. 7-5, 6-1.
Osaka’s only other quarterfinal appearance in the last year was at an Australian Open warmup tournament in Melbourne in January.
The former world No 1 is through to the third round at the Miami Open, easily handling 13th-seeded Angelique Kerber of Germany 6-2, 6-3 on Thursday.
Osaka trailed in the first set when a woman in the stands shouted, “Osaka, you suck.”
Osaka, the 2018 winner at Indian Wells, is back in the desert for the first time since 2019.