Williamson was due to fly home to New Zealand on Wednesday after suffering the injury while fielding during the 43-run Duckworth-Lewis method defeat in the opening one-day international.
The 23-year-old was unable to bat in the Black Caps' reply with a scan confirming a small fracture in his left thumb.
"This kind of injury unfortunately means Kane cannot take part in the rest of the tour and he'll return to New Zealand to begin his rehabilitation - Kane is about six weeks away from playing again at this stage," New Zealand physiotherapist Paul Close said in a statement.
Wicketkeeper-batsman Luke Ronchi will travel to Bangladesh to replace Williamson, with a an additional player set to be named "at a later date" for the upcoming tour of Sri Lanka.
The second of the three match ODI series will be played in Dhaka on Thursday.
The tour ends with a one-off Twenty20 international in Djaka on November 6.