Sammy suffered his injury when he was running around on a damp outfield in Queenstown before the third One-Day International (ODI) Jan 1. His place has been taken by left-arm spinner Nikita Miller.
"Darren is undergoing treatment and should be fit to return to playing duties in time for the regional Super50 tournament which starts later this month in Trinidad," said West Indies team physiotherapist CJ Clark.
ODI captain Dwayne Bravo will now lead the side in the T20 series.
The West Indies squad has been severely depleted in recent weeks as a result of injuries, resulting in only 12 fit players to choose from. Apart from Chris Gayle, who is recovering from a hamstring injury and injured Sammy, Marlon Samuels with a wrist injury and Darren Bravo have flown home while Ravi Rampaul suffered a broken finger, although he remains on the tour.
Kemar Roach and Kieran Pollard are also missing as the latter is recovering from a knee injury.
Three additional players - Andre Fletcher, Andre Russell and Samuel Badree - have joined the squad in New Zealand.
West Indies play their fifth and last ODI against New Zealand here Wednesday.