The British former world number one, 36, pulled out of events in Toronto and Cincinnati because of the issue.
Murray, ranked 37th in the world, has been practising fully at Flushing Meadows this week, with the final Grand Slam of the season starting on Monday.
“I think I made the right decision [to rest],” said the Scot, who faces France’s Corentin Moutet on Tuesday.
Earlier this month, Murray withdrew from the Canadian Open before his last-16 meeting with eventual champion Jannik Sinner.
The three-time major champion then decided not to play the following week in Cincinnati.
An initial assessment of the issue was inconclusive but further scans when he arrived in New York, which were analysed by Murray’s radiologist in the UK, confirmed he had a small tear.
“You have to be very careful with that,” said Murray, whose recent ranking of 36th was his highest since having the hip resurfacing surgery in 2019 which left him fearing for his career.
“You can’t protect that when you’re playing matches. You’re having to use that area a lot, especially when you’re serving.
“I had to take six or seven days off from serving and slowly build back up. It has not been perfect in that sense but my ab has been OK.
“The last few days have been good – no issues or any setbacks in the practices. So that’s really good.”