Mohammad Amir dragged Pakistan back into the game with career-best figures of five for 30 in 10 overs, but Australia still managed to post a challenging score of 307.
Kane Williamson's side have won all three of their matches to top the table ahead of their match against India in Nottingham on Thursday.
Gayle has hinted he could retire from one-day cricket after the World Cup and is eager to bow out on a high befitting his self-appointed "Universe Boss" moniker.
Stoinis sustained the injury during champions Australia's 36-run defeat by India at the Oval on Sunday.
The Proteas are battling to avoid their worst ever World Cup performance after losing their first three matches before Monday's washout against the West Indies.
The umpires' decision left tournament chiefs with the unwanted record for the most number of abandoned games at a World Cup, surpassing the two each at the 1992 and 2003 editions.
The 23-year-old, a nephew of Pakistan great Inzamam-ul-Haq, said the confidence he had gained from facing England quicks Jofra Archer and Mark Wood would help him tame the Australia attack.
South Africa's World Cup hopes are hanging by a slender thread after Monday's match against the West Indies was abandoned due to rain in Southampton.
Archer was born in Barbados to a British father and played for West Indies Under-19s before declaring his allegiance to England earlier this year.
Earlier, West Indies paceman Sheldon Cottrell reduced Faf du Plessis's team to 29 for two before rain stopped play.
Virat Kohli led India to their first-ever Test series triumph in Australia earlier this year to consolidate their position as the world's top side in the five-day format.
West Indies' pace attack of Oshane Thomas, Sheldon Cottrell, Andre Russell and Carlos Brathwaite have already made their presence felt at the World Cup, sparking memories of years gone by.
The opening batsman led the Indian charge after captain Virat Kohli won the toss and batted -- a decision he said was a "no-brainer" given the sunny conditions in south London.
New Zealand's third successive victory kept them on course to make the semi-finals in the 10-team tournament.
Following a surprise defeat by Pakistan, which followed England's opening 104-run rout of South Africa, captain Eoin Morgan urged the hosts to "stick to what we do well".