Down will be replaced by Georgia Plimmer, who plays for Wellington Blaze in the domestic circuit. The squad will also have Auckland quick Molly Penfold as a travelling reserve player.
"Quite simply, the entire squad is absolutely gutted for Lauren," head coach Bob Carter said on Saturday.
"She's a very popular member of the group and it's fair to say the team were quite emotional when we got the news she was out of the tournament."
Down suffered the injury while going for a catch in the fifth ODI against India. She will now fly home to seek medical advice.
"You saw the impact Lauren had in the recent series against India; she played some really mature innings in the middle order and her fielding was at an exceptional standard," Carter said.
"We wish her all the best with her recovery and know she'll be supporting us throughout the tournament."
New Zealand will open their World Cup campaign against West Indies on March 4.