Hungary's Barnabas Varga has surgery and can leave hospital on Wednesday
He suffered the injury when he clashed with Scotland goalkeeper Angus Gunn in the box after 68 minutes and appeared to be knocked out, needing urgent medical attention.
He was taken to hospital and was in a stable condition before his surgery on Monday.
"According to the information of the institution's doctors specialising in facial injuries, the intervention was uncomplicated, so the attacker ... can leave the hospital on Wednesday," the Hungarian football federation said on X.
Varga's girlfriend shared a picture of her holding his hand in hospital with the message: "Thank you for the kind, supportive messages. Barni is over the surgery. A little rest awaits him, but everything will be fine".
Hungary captain Dominik Szoboszlai had criticised medical staff for the length of time it took for Varga to get treatment on the pitch during the Group A game.
Varga will not play any part in the rest of Hungary's campaign, should they qualify for the knockout stage.
The win over Scotland ensured Hungary finished third in their group, but they must wait on other results to see if they qualify as one of the four best third-place finishers.