The world number 12 was trailing 6-2 2-1 when calling time on the match against Andreescu in a highly anticipated showdown between two of the past three US Open winners.
Raducanu initially left the court for treatment ahead of the seventh game of the opening set for a medical timeout and completed just four more games.
Speaking after her withdrawal, coming two weeks before the French Open, Raducanu explained she did not want to aggravate an injury sustained at last week's Madrid Open.
"I thought maybe taking one, two days off, it would go away because a lot of the other small niggles I've had, they've kind of gone away after taking, like, two days off," she said.
"Then I got here and I was training, but it just didn't seem to get better. I was training with some limitations. I wasn't moving really.
"I was just playing where I knew where the ball was coming, just staying in one corner. I think I must have underestimated the unpredictability of competition in a match.
"The last few weeks have been really positive. I've learned a lot about myself and my game has definitely improved on this surface but I need to make sure my back is fully right.
"I need to just keep on it. I don't want to play my next match with a feeling of limitation because I think that I learned my lesson from this week, when to push, when not to push."
Ons Jabeur joined Andreescu in the last 32 with a 6-0 7-6 (7-1) victory over Sorana Cirstea to keep her impressive run of form going.
The Tunisian won her first WTA 1000 title in Madrid last week and has now won seven successive main draw matches for the first time in her career
Elsewhere in Tuesday's action, American qualifier Lauren Davis pulled off a shock 6-2 6-3 win against number 11 seed Jelena Ostapenko.
Fellow Americans Jessica Pegula, Madison Brengle and Amanda Anisimova also advanced, beating Liudmila Samsonova, Marta Kostyuk and Tereza Martincova respectively.
Anisimova, who is now the player with the most wins in three sets so far in 2022 with nine, will face Belinda Bencic for a place in the last 16.