Varga collapsed to the pitch in Stuttgart and was struggling to move after chasing a long cross into the area as Scotland goalkeeper Angus Gunn came to collect the ball.
After a long delay, medics took the 29-year-old player from the pitch while red sheets were draped around the stretcher.
Hungary captain Dominik Szoboszlai appeared to be in tears as he consoled his stricken teammate.
Varga was immediately taken to hospital after the incident.
"The condition of Barnabas Varga is stable," the Hungarian Football Federation (MLSZ) wrote on X.
"The Hungarian national team player is currently in one of the hospitals in Stuttgart. We will inform you immediately if there is any news on his status."
German broadcaster Magenta TV reported that Varga was conscious and talking when he arrived at hospital in Stuttgart.
Hungary went on to win the Group A game 1-0 to stay in contention for the last 16.
