The Yankees added the procedure "went as expected."
The team confirmed last week Gregorius would need to go under the knife to repair a tear in his right ulnar collateral ligament.
Yankees manager Aaron Boone indicated the 28-year-old is at least a doubt to start the 2019 regular season.
"We're optimistic that he'll be back at some point during the season," Boone said. "I don't want to speculate too much on when. We think there's a realistic chance he plays the bulk of the season with us."
Yankees general manager Brian Cashman added that Gregorius' return date "all depends on how things play out."
The announcement put a damper on what was a career year for Gregorius. He set personal bests in home runs (27), on-base percentage (.335) and slugging percentage (.494). His 4.6 WAR ranked eighth among shortstops, per FanGraphs.
For some, Gregorius' elbow injury added further incentive for the Yankees to sign Manny Machado, who's a free agent this winer. MLB.com's Anthony Castrovince made some arguments against the move, citing the likely length of Machado's contract—potentially 10 years or more—and New York's need for starting pitching above a middle infielder.
The Yankees already had some motivation to spend big in free agency after a disappointing playoff exit in the American League Division Series. Now that Gregorius is officially on the road to recovery, the team will have another reason to evaluate all of its options in the free-agent market.