Desmond has been trying to play through the injury, but an MRI on Sunday revealed the tear.
Johnson said that Desmond was hurting at the end of Saturday's doubleheader with the Atlanta Braves.
"There was no point (Saturday) where I felt I made it worse," Desmond said. "I was just ready for it to stop."
Desmond, who missed the All-Star Game because of an abdominal injury, is hitting .286 with 17 home runs and 53 RBIs.
The Nationals lead the NL East by 3½ games after beating the Braves 9-2 on Sunday in the finale of the series between the top two teams in the division.
Information from The Associated Press was used in this report.