Teixeira sustained the injury Tuesday during batting practice with the Team USA World Baseball Classic squad as they readied for an exhibition game against the Chicago White Sox in Glendale, Ariz.
The Yankees slugger was initially diagnosed with a strained forearm and was expected to miss just 7-to-10 days.
But after an evaluation by Yankees team physician Dr. Christopher Ahmad and specialist Dr. Melvin Rosenwasser, Teixeira was found to have an ECU tendon sprain and was prescribed four weeks of complete rest, MLB.com reported.
The injury means he won't be rejoining the team during Spring Training in Florida and will be out of the New York lineup until mid-May.
"It's not what you want, but it's what you've got to deal with," Girardi said. "It's life."
The Yankees last month also lost slugging outfielder Curtis Granderson for 10 weeks with a fractured right forearm.