New Zealand had earlier said that Williams would miss the three-match home series after aggravating an existing injury while training with his Super Rugby team the Auckland Blues this week.
The All Blacks said Williams' keyhole surgery to tidy up a loose piece of bone in the knee would keep him out of action for at least four weeks.
"It was only a small op but the doc said it was successful," the 32-year-old Williams wrote on his verified Twitter account on Friday.
"I'm going to miss running out there with the Blues/All Blacks. However, I'm genuinely excited to watch my fellow midfield brothers in that black jersey showcase their amazing skills.
"I'm excited to help the team where I can but for now better help my wife with these nappies," he added alongside a picture of him in hospital with his children
Williams' absence is a concern for All Blacks coach Steve Hansen, with Canterbury Crusader Ryan Crotty still battling concussion symptoms after a head clash against the Blues.
Crotty will miss the Waikato Chiefs game on Saturday, with the All Blacks assembling in Auckland the day after ahead of the first test at Eden Park on June 9.
Williams and Crotty have been Hansen's first-choice centre pairing since the 2015 Rugby World Cup.
The potential absence of Crotty, who can play both midfield positions, could ensure a lengthy run in the number 12 jersey for the Wellington Hurricanes' Ngani Laumape.
The 25-year-old is likely to partner either Anton Lienert-Brown or Jack Goodhue in the midfield.
Goodhue, 22, and the 23-year-old Lienert-Brown can also play both midfield positions. All three have been tipped for major roles in Hansen's 2019 World Cup plans.
Williams fractured his wrist in March and only returned to the field for the Blues three weeks ago against the Hurricanes.
He had been named to start against the Melbourne Rebels at Eden Park on Saturday but his expected absence could mean he might not even be fit in time for the Blues' next Super Rugby match against the Queensland Reds on June 29.
(Reporting by Greg Stutchbury; Editing by Ian Ransom/John O'Brien)
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