The 22-year-old is out of the match with a side strain.
Milne missed the final group game against Bangladesh after injuring a shoulder against Afghanistan but recovered to face West Indies in Saturday's quarter-final win.
Milne felt discomfort yesterday following the quarter-final against the West Indies and underwent an MRI scan yesterday afternoon. The results showed significant swelling around the region and he will not be fit to play any further part in the tournament.
A timeline for his return is not yet known.
Milne has taken five wickets at the World Cup so far. Either Mitchell McClenaghan or Kyle Mills will likely replace him.
McClenaghan came in for Milne in the Bangladesh game and struggled, taking none for 68 from eight overs. Mills has yet to play at the tournament.
