Pavard was involved in a collision with Germany’s Robin Gosens in the second half of Tuesday night’s match in Munich.
He said afterwards he had been “a little knocked out for 10 or 15 seconds” and hit the ground after the incident without getting his hands out to cushion his fall.
Concussion substitutes were not approved for the tournament, with UEFA instead trialling them at its European Under-21 Championship.
Peter McCabe, the chief executive of Headway, said: “The way this incident was handled was sickening to watch.
“UEFA has to come out and immediately explain how it was allowed to happen and what action it will now take to ensure something similar does not occur in the future.”
Headway said Pavard had received less than three minutes of treatment before coming back into the action.
“This is another example of football authorities failing to protect the short- and long-term health of a player,” McCabe added.
“It was plain for all to see that Pavard was unable to protect himself from the fall. Pavard’s later statement that he lost consciousness confirms the seriousness of the incident.
“We have continuously been told that football’s concussion protocols are fit for purpose and that temporary concussion substitutes are not necessary.
“But here we have yet another example where it is simply not credible to suggest that a concussion could not be ‘suspected’ or a possible consequence of the impact. However, after a brief on-pitch assessment the player was allowed to continue.
“Furthermore, it appeared that the referee was attempting to speed up the medical team and usher them and the player off the pitch, rather than allowing them the time they needed to assess the seriousness of the injury.
“Why is it so hard for football to accept it has got this wrong and follow the example set by other sports by introducing temporary concussion substitutes?”
Uefa has been contacted for comment.
World players’ union FIFPRO and the English union, the Professional Footballers’ Association, wrote to the game’s law-making body the International Football Association Board in April asking it to extend the scope of its trials to cover temporary concussion substitutes.
The letter warned that the current protocols had “jeopardised players’ health”.
The Premier League and the Football Association trialled permanent concussion substitutes in its competitions from February.