Szoboszlai was among Hungarian players waving frantically for help as Varga, who fractured several bones in his face, lay motionless on the ground after a collision with Scottish keeper Angus Gunn 68 minutes into the game.
When help finally arrived, Szoboszlai was seen grabbing the stretcher to try to hurry the medics along.
Varga was wheeled away for treatment, concealed behind a large brown cover held by officials.
"I don't understand this kind of protocol, why the medical staff don't rush in at this time?" Szoboszlai told Hungarian reporters.
"This question remains for everyone, so I hope that everyone can save a few seconds and save a life," said Szoboszlai, who was briefly in tears over the collision.
"I hope we will never have to experience this again ... something needs to be done."
Coach Marco Rossi confirmed Varga regained consciousness before leaving the field and was taken to hospital where his condition was stable.
Hungary must wait for the outcome of other group matches to see if they will advance to the next round as one of four best third-place finishers.
The Varga incident seemed to spur Hungary on after what had been a dismal game with few chances, Szoboszlai's team stepping up a gear with a flurry of late opportunities that led to Kevin Csoboth's breakaway goal in the 100th minute after a lengthy period of stoppage time.
Szoboszlai celebrated the victory by wearing Varga's spare shirt turned back to front.