Milne, 20, missed the T20s in December after suffering an Achilles problem while warming up for a domestic match and although he was included for the ODI leg of New Zealand's tour, the problem as flared up again.
New Zealand physiotherapist Paul Close said: "Adam sustained a recurrence of his lower leg injury that's been problematic for him over the last few months. He requires further assessment to determine the severity of the injury but because he is unable bowl without pain the decision has been made to replace him for the ODI series."
Wagner, who qualified to play for New Zealand last year after moving from South Africa, has played two Tests but is yet to feature in either of the limited-overs teams. He could make his ODI debut in the three-match series, beginning on January 19.
Mike Hesson, New Zealand's head coach, said: "While it is obviously disappointing for Adam this gives Neil a wonderful opportunity to join the squad for the ODI series. He's the ideal replacement for Adam and we are confident he will perform well in the one-dayers should he get the opportunity on this tour."
Earlier this week, New Zealand were forced to replace allrounder Corey Anderson with Grant Elliott, after Anderson suffered a fractured thumb in training.