The Wellington seamer will be replaced by Canterbury quick Kyle Jamieson.
Milne suffered the injury while bowling in the SA20 on Sunday, with subsequent scans revealing the extent of the injury.
Jamieson, who is currently part of the Black Caps white-ball tour of India, makes the step up after originally being named as the travelling reserve for the February-March tournament in India and Sri Lanka.
Black Caps coach Rob Walter was empathetic towards Milne.
"We’re all gutted for Adam," Walter said.
"He’d worked so hard to get himself ready for the tournament and was looking back to his best in his eight games for the Eastern Cape Sunrisers.
"It's unfortunate timing for Adam and we wish him a speedy recovery."
Walter said Jamieson was a worthy replacement.
"It’s great that Kyle is already with us here in India.
"He’s an integral member of our pace-bowling group and has hit the ground running on this tour.
"He’s a hard worker with a good set of skills and experiences that will hold him good stead for the tournament."
Walter confirmed a replacement travelling reserve for the T20 World Cup would be confirmed in due course.
— Allied Media
