Johnson was put on the disabled list April 26. He had an operation Saturday.
Manager Matt Williams said Monday that Johnson is resting at home, but could not begin to guess at a timetable for his return.
Asked if he expected Johnson to play again this season, Williams replied: "I don't know. It's a question of rehab and how it comes along. That one is up in the air."
Johnson batted .222 in 12 games this season, going 1 for 7 as a pinch hitter.
The 38-year-old broke into the majors in 2003 with Toronto and has played for six teams. This is his first year with Washington.