After spending nearly seven months on the sidelines with a knee injury, Williamson made 78 not out in his return to action in the team's comprehensive victory against Bangladesh on Friday.
Williamson, however, was struck by a throw while running between the wickets during the match, which now makes him uncertain for the tournament.
Tom Blundell will travel to India as cover, but Williamson will remain with the team with the hope of being available for the back end of group play next month, New Zealand Cricket said in a statement on Saturday.
"Firstly, we're all feeling for Kane to have this occur after all his hard work to return from his knee injury," Stead said.
"While it's disappointing news, the initial diagnosis has given us some optimism he can still feature later in the pool play following a period of rest and rehabilitation.
"Kane is clearly a massive part of our side and a world-class player and captain - so we'll look to give him every opportunity we can to return in the tournament."
Blundell was picked ahead of Henry Nicholls, who suffered a mild side strain during a Plunket Shield warm-up match the week.
"Tom's been with the one-day side on tours to Pakistan and Bangladesh and performed well," Stead said.
"He covers multiple positions in the batting order and his wicket-keeping skills are also an added bonus as a back-up."
The 2019 runners-up have kicked off their campaign for a maiden World Cup title with a hat-trick of victories. They face Afghanistan in their next match in Chennai on Wednesday.
