Cricket South Africa (CSA) announced on Monday that the three-match Test series, which begins on Sunday in Centurion, won't admit any spectators.
The teams are then scheduled to play three ODIs starting from January 19.
The South African board said in a statement that the decision to play behind closed doors had been taken jointly with the Indian cricket board (BCCI).
"This decision was taken in order to avoid any breaches that could compromise the tour from a COVID-risk perspective and to also maintain a hazard-free bubble environment," the statement read.
The statement added that the decision was made in the best interests of the game as well as the health and safety of players and staff.
Earlier on Monday, it was also announced that South Africa's Twenty20 Mzansi Super League had been cancelled for the second time due to COVID-19 related issues.
The 2022 Mzansi Super League was scheduled to be held in February but the board said increased travel restrictions on South Africa after the emergence of the Omicron variant made it difficult to host the tournament.
The 2020 edition was also cancelled due to logistical challenges amid the pandemic.
The board said the Mzansi Super League would be replaced by a domestic T20 Challenge tournament featuring eight first division teams.
South Africa reported a record number of daily COVID-19 infections last week in a fourth wave.
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