Uncapped hooker Bradley Slater has been called up to the squad to replace Bell, who was injured on Sunday and is expected to be sidelined for six to eight weeks.
Slater will arrive in Cape Town on Wednesday so is unlikely to face the Stormers on Friday.
"We are all gutted for George, to be ruled out of the tour so early," All Blacks coach Dave Rennie said in a statement.
"He's had a frustrating past few months with injury so it's a real blow for him to have injured his calf again."
Rennie already had three other hookers in his bumper squad in Codie Taylor, Samisoni Taukei'aho and Asafo Aumua.
The All Blacks will face provincial opposition in the shape of the Stormers, Sharks, Bulls and Lions as well as four tests against the world champion Springboks on the tour.