Hernandez, who kicked two penalties in Argentina's 27-6 defeat by the Springboks in their tournament debut last weekend, suffered the injury near the end of an intense training session in Buenos Aires on Tuesday.
Argentina, playing in the southern hemisphere's elite championship for the first time, host South Africa at the Malvinas Argentinas stadium in the Andean city of Mendoza in the west of the country.
The loss of the injury-plagued former Natal Sharks fly-half Hernandez is a blow to Pumas coach Santiago Phelan. He had named a team with just one change from the opening match at Newlands in Cape Town, bringing in Martin Rodriguez at full-back in place of Lucas Gonzalez Amorosino.
Phelan will announce a replacement for Hernandez after Thursday's training run in Mendoza.
Nicolas Sanchez is a reserve fly-half but Phelan my opt to reshuffle his back line with the more experienced Santiago Fernandez standing in for Hernandez.