Later on Wednesday, the Lions announced two players were isolating after one of the management team tested positive.
The Lions' warm-up fixture against the Bulls on Saturday was postponed because of Covid-19 cases in the hosts' camp.
South Africa is currently battling a third wave of infections.
The Springboks have had to isolate twice in the past two weeks, with their three-Test series against the Lions taking place on consecutive weekends from 24 July.
There have been 12 positive tests among South Africa's players and staff this week and six in the Georgia camp.
"In the context of the loss of life and economic damage that Covid and this third wave are wreaking, the cancellation of a rugby match is pretty trivial," said SA Rugby chief executive Jurie Roux.
"We continue to plan for the Springboks' re-emergence from isolation and the completion of the Test series - but in the short-term, we wish a speedy recovery for those who have been infected."
An SA Rugby statement said the squad's travel plans are being reviewed "to comply with Covid-19 protocols", with the Lions set to face a South Africa A team in Cape Town on 14 July.
The Bulls game is the first to be affected by coronavirus on the Lions tour and organisers said they hoped to reschedule the match or find other opponents.
On Wednesday, the Lions were retested, and their game against Cell C Sharks will go ahead if no further positive results are returned, albeit with a later kick-off time of 19:00 BST.
The host nation previously suspended training on 27 June because of three positive tests, but were later cleared to play.
Second row Lood de Jager earlier tested positive, with Vincent Koch, Herschel Jantjies and Sbu Nkosi having returned positive results on 27 June.
Jantjies later returned a negative test and was a replacement when the Springboks beat Georgia 40-9 last Friday in Pretoria - their first Test match since winning the 2019 World Cup.
The following day, the Lions played their first game in South Africa, beating the Sigma Lions 56-14 in Johannesburg.
Speaking before the game was called off, Lions defence coach Steve Tandy said the tourists would adapt to any disruption.