Henry sustained the injury during Wednesday's match against South Africa in Pune, with a scan confirming a lower tear which will require up to four weeks' recovery, the team said on Friday.
"We’re gutted for him," coach Gary Stead said in a statement.
"Matt's been a crucial part of our one-day side for a long time and to see him ruled out as we reach the business end of this tournament is immensely disappointing."
The 6-foot-8-inch tall Jamieson arrived in Bengaluru late on Thursday and was expected to train with the team later on Friday.
Stead said he would be ready for Saturday's match against Pakistan if required.
"We’re fortunate to have a player of the class of Kyle waiting in the wings," he said.
"Kyle’s had to work really hard to return from two separate back injuries and I know he’s really excited about being involved in his first ODI World Cup."
Question marks remain over the fitness of Jimmy Neesham, who suffered a bruised right wrist after being hit by the ball while bowling in the defeat by South Africa although the player did not break any bones.
Mark Chapman (calf), regular skipper Kane Williamson (thumb) and paceman Lockie Ferguson (Achilles) are nursing injuries but batting all-rounder Daryl Mitchell backed the squad to rise to the challenge.
"We've obviously got training this afternoon and a number of guys have got to get through fitness tests and pass them and we probably won't know till tonight and tomorrow morning how they pull up," Mitchell told reporters.
"But yeah for us, we back everyone within our squad to have success and do a job for the team.
"We're all pretty clear on the roles we're going to play if given the opportunities and we'll just go out there get stuck in as Kiwis do and hope we come around the right side."
The Black Caps sit fourth in the standings with two matches in the group phase remaining against Pakistan on Saturday and Sri Lanka next week.
"All I know is that the 11 guys that will take the field tomorrow will be very proud to represent our country and get stuck in a World Cup," Mitchell added.
"We won't make it bigger than what it is. It's another two points for this tournament, which is important for us in the big scheme of things, but we'll keep playing like Black Caps do and I'm sure we'll come a long way to winning the game."
