The 22-year-old had been suffering pain in his left heel for the last two weeks, coach Mike Hesson said.
While he was pain-free before the quarter-final against West Indies, it had 'flared up again'.
"It's to a point now where he wasn't able to bowl so he's unable to bowl and he's in a lot of pain when he bowls," Hesson told reporters at Eden Park.
"He's fine walking in a straight line but unfortunately when he bowls it's become increasingly uncomfortable to a point where he's unable to carry on."
Milne's injury has created a selection dilemma, with virtually the same side picked for all of New Zealand's World Cup matches.
Hesson was loathe to confirm whether Henry, whose inclusion was yet to be ratified by the governing International Cricket Council, would come into the side as a 'like-for-like' pick.
"Once we get to the ground tomorrow we'll look at whoever is available in the squad and we'll look at the wicket again and we'll pick what we deem to be the best squad tomorrow," he said.
Swing bowler Kyle Mills and left-armer Mitchell McClenaghan are also available, though Mills often opens the bowling and it would be a difficult decision to split the new ball pairing of Trent Boult and Tim Southee.
Unlike Mills, McClenaghan bowled in the tournament when he replaced Milne in the final pool game against Bangladesh but he struggled for line and length and was expensive.
The 23-year-old Henry has been playing steady cricket in New Zealand's first-class competition.
"It's certainly something that's in his advantage but we've got plenty of options," Hesson said.
(Editing by Mark Meadows / Ian Ransom)
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