Before scheduling surgery, Crow, 28, will get a second medical opinion.
"What the doctor told me is my UCL [ulnar collateral ligament] is partially torn," Crow said.
The reliever also said Tommy John surgery was recommended.
"I'm still trying to figure that out and get a second opinion," he said. "If it's the same thing, I'll probably get surgery. I'll just wait for that."
The Marlins acquired Crow from the Royals in November. The right-hander appeared in seven Grapefruit League games and logged eight innings, striking out nine and walking four. He posted a 2.25 ERA.
"It's too bad," manager Mike Redmond said. "He was fighting to make our ballclub. That's a tough break, not only for us, but for him, to get hurt and not be able to have a chance to make the team."
Crow pitched a scoreless inning on Saturday, and his previous appearance before that was last Wednesday, when he tossed an inning against the Tigers in Lakeland, Fla.
Everything felt normal until Sunday morning.
"The last time I pitched was Saturday," Crow said. "I felt fine then. But I came in the next day [and] it was really sore and it was painful to throw. Everything before that was pretty much normal, like normal soreness. It got really bad on Sunday."
Crow, signed for $1.975 million this season, is eligible for free agency in 2017. His injury opens the door for right-hander Sam Dyson to secure the final bullpen spot.
Joe Frisaro is a reporter for MLB.com. He writes a blog, called The Fish Pond. Follow him on Twitter @JoeFrisaro. This story was not subject to the approval of Major League Baseball or its clubs.