Cabrera underwent a fluoro exam, which was negative, and he will have a MRI examination on Wednesday morning after being diagnosed with a right shoulder strain.
Yankees manager Joe Girardi said that he would be "completely shocked" if Cabrera was available to play Wednesday as New York completes a three-game series in Texas.
Cabrera was hurt while chasing Ian Kinsler's deep drive to center field leading off the frame for the Rangers against Yanks starter Joba Chamberlain.
The 24-year-old could not catch up to the ball and crashed into the center-field fence, immediately grabbing for his right shoulder and losing his glove before picking the ball up and throwing it in.
"I hit the wall with my forearm and then my shoulder went way back," Cabrera said.
Cabrera then staggered around the warning track in obvious pain, drawing Girardi and head trainer Gene Monahan all the way to the outfield before Brett Gardner replaced him.
"I'm sure he's going to be sore," Girardi said. "The fact that the [tests] came back negative is good. But he was sore to push on it when Geno examined him, and that's a concern."
Cabrera said that he had treatment during New York's 7-3 loss to the Rangers and was optimistic he might avoid the disabled list.
"I feel better right now because I did ice three times and a couple of exercises," Cabrera said through an interpreter. "I think it might be a couple of days."
Cabrera lost a spring battle to serve as New York's Opening Day center fielder but soon regained the job from Gardner, batting .323 with five home runs and 20 RBIs entering Tuesday's play.
The switch-hitter had hit safely in 29 of his past 32 starts, batting .333 (42-for-126) over that stretch. The start of Tuesday's game was delayed two hours and 24 minutes by rain.
Bryan Hoch is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.