Gordon suffered the injury in the 70th minute of the Mendoza clash, in which Australia were unable to secure a winning score despite Argentina being reduced to 13 men for the final 10 minutes.
The playmaker, who had emerged as Kiss's first choice for a position the Wallabies have struggled to fill for the last few years, will miss the meeting with world champions South Africa in Perth on September 27.
Australia also play home-and-away tests against New Zealand in mid-October before heading to Europe for the resumption of the Nations Championship in November.
Gordon is scheduled to get married on his return to Australia this week, the Nine media report said.
The 25-year-old's career has frequently been interrupted by injury.
He missed the Nations Championship tests against France and Italy in July because of a calf injury having struggled with leg issues throughout the Super Rugby Pacific season.
Gordon returned to rugby union late last year after a season in rugby league with the Gold Coast Titans. He was restricted to just one game in the National Rugby League by a rare spinal condition.
Australia are unbeaten in four tests since Kiss took over from Joe Schmidt after the July internationals.