He had travelled back to his home town of Tauranga to attend a midwife appointment and missed today's training session.
It initially appeared that Williamson would be ready for tomorrow's test after he returned to Wellington to rejoin the test squad, with coach Gary Stead saying he was confident his skipper would play.
However, Williamson has changed his mind and has decided to travel back to Tauranga tonight to be with his wife.
Speaking to Newstalk ZB's Heather du Plessis-Allan, Black Caps head coach Gary Stead confirmed that Williamson had a late change of heart.
"I guess just the closeness to the birth time and just needing to be there in terms of family support. Family comes first. Kane's not the first player to miss a test match because of an impending birth and he won't be the last," Stead said.
"We dont know 100 per cent for sure yet [when the baby will arrive], and I am no medical expert but there's a good chance that something will happen in the next six to 10 days."
Stead said that Will Young, who made his test debut in the Eden Park clash, will slot into the top order in Wellington.
"It's always disappointing when you lose a person of Kane's calibre and class as a person but you always plan for contingencies things like this.
"Will Young will come into the team and bat at number three, and that is a natural position for him as well and a great opportunity for Will.
"We certainly aren't taking the West Indies lightly. It was a really clinical performance last time but I'm hoping we can produce something similar."
The loss of Williamson will be a big blow for the Black Caps as they hope to get their second test win over the West Indies.
Williamson scored 251, his third double century in test cricket, as part of a winning performance by the Black Caps in the first match of the West Indies series and looks in fine form ahead of tours against Pakistan and Australia over the remainder of the summer.