The accident occurred when Navarro slipped on the top step heading into the Rays' dugout after the second inning of Friday night's 13-4 Rays victory.
"I reached back, I was trying not to break my neck, and I caught the rope from the netting," Navarro said. "I didn't say nothing, I just showed my hand. ... I almost threw up when I saw it."
Rays manager Joe Maddon said he couldn't look at the cut.
"It was really an ugly cut," Maddon said. "We're thinking [he'll be out] two weeks."
Navarro cut the ring and middle fingers of his right hand and was taken to Columbia Presbyterian Hospital where he was seen by Dr. William Levine. The cut on Navarro's ring finger required eight stitches, and the middle finger wound took four.
Mike DiFelice was selected from Triple-A Durham to take Navarro's place. The veteran receiver showed up at Yankee Stadium two hours before the start of Saturday afternoon's game against the Yankees.
DiFelice was pulled out of Friday night's Bulls game at Toledo and told he was on hold.
"I only had one at-bat [when he got removed], and I'm like, 'This guy's a tough manager,'" DiFelice said. "I struck out, but I had some good swings. I haven't seen live pitching in a week. I was on hold. Just wait."
DiFelice caught a 6:30 flight Saturday morning from Toledo to Cincinnati en route to New York.
Injuries are "part of the game," DiFelice said. "You never want to see anyone get hurt, any player. ... I'll do what I can to help the team."
Bill Chastain is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.