The forward was hurt in a clash of heads with Inter defender Milan Skriniar as the Nerazzurri ended leader Napoli's unbeaten start to the Serie A season on Monday (AEDT).
He was substituted in the 55th minute and taken for tests, which the club later confirmed had revealed "multiple displaced fractures of the left orbit and cheekbone".
Osimhen had an operation on Wednesday that included the fitting of titanium plates and screws.
The club confirmed he will be out of action for three months, meaning he will play no part in Nigeria's AFCON campaign in Cameroon, with the tournament beginning on 10 January.
Osimhen is Napoli's leading goalscorer in 2021-2022, with five in Serie A and four in the Europa League.
He scored seven times in September alone, a tally matched only by Real Madrid striker Karim Benzema across Europe's top five leagues.
Only Borussia Dortmund's Erling Haaland was younger than Osimhen, who turns 23 next month, among players in the top five leagues to score 10 or more goals in both 2019-2020 and 2020-2021.