Stander, playing against the country of his birth, was dismissed after 20 minutes for a late tackle that left South Africa first five-eighth Pat Lambie concussed.
Despite going down to 14 men, Ireland went on to win 26-20, their first victory over the Springboks.
After a lengthy hearing on Monday, judicial officer Terry Willis of Australia dismissed the notion that the referee's decision to send Stander off was wrong.