The two sides were scheduled to travel together to Johannesburg, with West Indies continuing on to Antigua.
The charter had been arranged after both teams were stuck in India following the end of the ICC Men's T20 World Cup, as tensions in West Asia disrupted international air routes.
West Indies have been in Kolkata since their Super Eight defeat to India at Eden Gardens on March 1, while South Africa have remained there since their semifinal loss at the same venue on March 4.
Reports say 12 West Indies players and 10 support staff, along with 12 South Africa players and two staff members, are still in the city.
With the charter flight cancelled, several players have begun arranging individual travel plans, though these depend on airspace approvals.
The International Cricket Council and Cricket West Indies are working to find alternative travel options, but no confirmed departure date has been announced.
