Hungary’s players quickly surrounded their team-mate and a large screen was brought out to provide some privacy as medical staff treated him.
Players on both sides were seen with their heads in their hands, sparking memories of Christian Eriksen’s cardiac arrest at the last Euros tournament in 2021.
Varga, a forward, was stretchered off to warm applause from all sides of the stadium, with his colleagues continuing to surround him, while holding up the screen.
A statement from the Hungarian Football Federation later read: “Barnabas Varga’s condition is stable. The Ferencvaros player is currently in one of the hospitals in Stuttgart. We will inform you immediately if there is any news about his status.”
And Hungary head coach Marco Rossi added after the match Varga would be “okay” but may need an operation on his cheekbone.
Soon after, Scotland thought they had a penalty when substitute Stuart Armstrong was brought down in the penalty area, but the referee waved away their appeals.
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Scotland captain Andy Robertson said they were caught by a “sucker punch” at the end of the game in Stuttgart.
Speaking to the BBC after the match, Robertson said they “gave it everything” and “it could’ve gone either way”.
The Liverpool full-back reflected: “That’s football, that’s how it goes. It’ll take a long time to get over this.”
His Reds team-mate, Dominik Szoboszlai, the Hungarian captain, brought a save from Gunn in the Scotland goal late on – only to find out he was offside.
Earlier, he had been in tears as medical staff responded to his teammate, according to TV commentary of the game.
Just moments after the Szoboszlai chance, Hungary’s Kevin Csoboth hit the post when clean through.
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1:30
‘Devastated’ Scotland fans ‘blaming manager’ – Sky’s Rob Harris reports
The frantic end was in sharp contrast to a cagey opening period, with few, if any, clear-cut chances.
In the group’s other game, Switzerland drew 1-1 with hosts Germany, after leading for most of the match.
The draw means Germany top the group with seven points, with Switzerland second on five. Hungary are third on three points, with Scotland bottom on one.
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Scotland fans leaving the ground told Sky News the team was “too negative”, with several blaming manager Steve Clarke and another accusing the team of “playing for a draw when we needed a win”.
One man in the Tartan Army’s standard outfit of replica shirt and kilt simply said it had been “a nightmare, an absolute nightmare”.
Sky’s Rob Harris, in Stuttgart, said: “When it came to the crunch, they Scotland just didn’t have enough and barely threatened the Hungary goal.
“When they did push forward, it was to their cost, because that’s when Hungary claimed their winner.”
The result follows a 1-1 draw in Scotland’s second game of the tournament, against Switzerland.
Thousands of supporters have travelled to Germany to support Scotland’s Euros run.
Scotland played in Euro 2020 in the summer of 2021, which was delayed by the COVID pandemic. But this is the first major football tournament without social distancing concerns for Scots since the World Cup in France in 1998.
It was the 12th time Scotland have reached the finals of a major tournament but failed to make it through to the knockout stage.