Schar collided with Georgia’s Jemal Tabidze and received emergency help.
The incident occurred after 24 minutes and Newcastle United defender Schar went on to complete the game.
Headway chief executive Peter McCabe said: “What is it going to take to make football take concussion seriously?”
Schar lay unconscious on the ground after the clash of heads with Tabidze during the match in Tbilisi and Georgian player Jano Ananidze rushed to his aid.
The 27-year-old recovered quickly after further treatment from SWISs first-aiders, SWISs newspaper Blick reported, and was able to continue playing, helping set up Switzerland’s second-half goals for their 2-0 victory.
“It looks awful. I can’t remember anything,” the Newcastle player told Blick after being shown video footage.
“I was out for a few seconds. My skull is still humming. And I’ve got neck ache and a bruise on my forehead. But it was worth it.”
Tabidze also lay motionless after the clash, his shirt covered in blood, but he, too, recovered.
McCabe said: “How many more players will have their careers and, more importantly, their lives and long-term health put at risk by the sport’s inability to follow its own protocols?
“Put simply, the decision to allow Fabian Schar to return to the field of play after suffering a clear concussion was not only incredibly dangerous, but also a clear dereliction of duty.
“The player’s comments after the match are also deeply disturbing and again show the lack of awareness and understanding among players.
“UEFA must immediately launch an investigation into the incident and explain why their protocols were not followed.”