Milne felt discomfort in his left foot while landing to bowl in training, and has been replaced by Seth Rance.
"Adam has a great work ethic and puts a lot into his fitness, so I certainly feel for him with this injury," said selector Gavin Larsen.
"It's a long summer and there's a lot of cricket left for Adam to play, so we'll work with him to get him back to full fitness."
Rance comes into the side after a solid start to the domestic season with Central Districts, having played two ODIs earlier this year.
"Seth comes into the squad off the back of consistently strong domestic performances," said Larsen.
"We were already looking at Seth for the upcoming Twenty20 series, so this just brings him into the squad slightly earlier than anticipated."
The Black Caps take on the West Indies in the second ODI in Christchurch on Saturday. They lead the series 1-0.
