Meyer underwent an MRI on Monday that revealed the injury, and the decision to have the surgery was made after he consulted with multiple doctors.
The surgery will be performed by Dr. Neal ElAttrache in Los Angeles on a date that has not yet been determined.
Also Thursday, Miami left-hander Daniel Castano had to leave the Marlins' game at Cincinnati in the first inning after being hit near his left eye by a line drive off the bat of Donovan Solano, the Reds' fifth batter. Castano walked off the field under his own power after being looked at by Miami medical personnel.
The Marlins announced that Castano has a forehead contusion and mild concussion and will be evaluated further.
Meyer, 23, is one of the top prospects in baseball and was ranked No. 33 overall before the season by ESPN's Kiley McDaniel.
Meyer faced three batters and threw only 10 pitches before being removed in the first inning of his start Saturday against the Pittsburgh Pirates. The Marlins had placed him on the 15-day injured list Sunday.
The third overall pick in the 2020 draft, Meyer made two starts this season before the injury and finishes with an 0-1 record and a 7.50 ERA in six innings. He allowed five runs, seven hits -- including two home runs -- struck out six and walked two.
The Associated Press contributed to this report.