The Kings announced Tuesday that Greene will be out indefinitely.
Greene has played in just three games this season for the first-place Kings, going scoreless with eight penalty minutes.
The 32-year-old former alternate captain was a key contributor to Los Angeles' Stanley Cup-winning teams in 2012 and 2014, providing physical defense and steady leadership.
He scored a career-high 15 points in 2011-12, but has scored just 16 points in 128 games over the past four seasons. Greene played five games during the 2012-13 regular season, undergoing surgery on a herniated disk in his back.
Greene again was limited to 38 games during the 2013-14 regular season, but played all 82 games last year.