The Hungarian striker, 29, clashed with the Scotland goalkeeper, who came out at a free-kick to punch the ball clear in their Euro 2024 match.
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However, the pair accidentally made heavy contact, leaving both men down.
Gunn got back to his feet quickly but Varga stayed down motionless in harrowing scenes as team-mates put him in the recovery position.
The forward was pushed to hospital in Stuttgart, where he was conscious and stable.
According to Varga’s father Andras, he underwent emergency facial surgery this morning after a visit from his club side Ferencvaros’ doctor Gergely Panics.
Panics gave an update on the footballer’s condition to the press – and detailed the injuries sustained.
He told Hungarian news outlet Magyar Nemzet: “Barni suffered a concussion during the collision and lost consciousness.
“By the time he reached the ambulance, he had already regained consciousness, after which he was taken to the clinic in Stuttgart, where, after the examination, a therapeutic plan was drawn up, according to which he would be operated on.
“Our striker suffered multiple facial skull fractures, several smaller and larger bones were broken in his face, and some were even dislocated.
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“After the operation, he will be kept in for two days. If everything goes well, he will be released on Wednesday.”
Varga collided with Gunn during the second half of the Group A finale on Sunday night.
Furious Hungarian players were angered by the time it took the walking medics to reach the stricken Varga and start treating him in the penalty area, with stewards holding up blankets to create a screen.
Captain Dominik Szoboszlai believes Uefa should make a change to ensure players in a similar situation receive treatment as quickly as possible.
The Liverpool midfielder said: “I don’t really know the protocols or how it works, but if our doctors say we need someone immediately to help, then I don’t think they should walk.
“It’s not my decision, but I think we need to change something.
“Even if someone is on the ground, and you know that it was a big collision, then just go on [the pitch].
“Even if the referee is saying don’t go on, just go on. And if you can see it’s nothing serious, then you can just go down and it’s finished. But seconds can help a lot.”
Panics refused to be drawn on the reaction of the medical team.
He added: “Regarding the German protocol and first aid, I do not want to comment, as I do not know it in detail and I am not responsible for it.
Inside Barnabas Varga’s rise from semi-pro to Euro 2024
BARBABAS VARGA has enjoyed a meteoric rise from semi-pro leagues of Austria and Hungary to the Euro 2024 finals.
The Hungary striker played for six seasons, between 2010 and 2016 for SV Eberau in the Bergenlandliga – the regional fourth tier of Austrian football.
A record of over a goal every other game earned Varga a shot at the big time with with SV Mattersburg in the Austrian Bundesliga.
However, he scored only once in just 16 games over the next three seasons before dropping down to the second tier with SV Lafnitz.
After a mixed spell, Varga’s next step was to his homeland, again in the second tier, with Gyirnot in 2020.
That’s where it began to click for Varga and his eye for goal saw top division Paksi FC swoop.
Varga scored almost a goal a game in his first season with Paksi, finishing as the top scorer in the Hungarian league.
That earned the late bloomer an international debut, at the age of 28, in a Euro 2024 qualifier against Bulgaria.
One his third cap, away to Serbia, he netted his first Hungary goal.
Last summer, Varga joined Hungarian giants Ferencvaros and again got the golden boot as his side won the league, the first silverware of his career.
“However, I can say that I had the opportunity to visit Barna in the hospital, and I saw that he is in expert hands and is receiving a lot of attention.”
Uefa, however, defended those who treated Varga and insisted all the guidance was followed as they reached him within 15 seconds.
The BBC were forced to apologise because the Uefa live feed showed a replay of the incident despite the severity of the incident.
Hungary players looked visibly distraught after witnessing their team-mate’s horror injury.
However, following a lengthy delay, the game carried on and Kevin Csoboth scored a 100th-minute winner to keep Hungary’s hopes of qualification for the knockout stages alive.
Players celebrated by holding up and wearing a ‘Varga 19’ shirt for their striker.
Varga, though, will play no further part in the tournament as he begins his recovery.
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