Milne suffered the injury while bowling in South Africa's SA20 league on Sunday, with scans later revealing it was bad enough to prevent him from playing in next month's World Cup in India and Sri Lanka.
Milne, 33, has been replaced by all-rounder Kyle Jamieson, who is playing for the New Zealand white-ball teams in India as they prepare for the World Cup.
"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."
Jamieson would be a worthy replacement, Walter said.
"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 [in] good stead for the tournament."
Walter confirmed a replacement travelling reserve for the T20 World Cup would be confirmed later.
The Black Caps are playing India in the second of five T20 internationals in Raipur on Saturday at 2.30am (NZ time).
