Playing for Italy, Gallinari hit the Netherlands' Jito Kok during a scuffle under the basket, causing an injury that sent him to the local hospital.
Gallinari will miss the Eurobasket tournament, but is expected to be fully recovered for the start of training camp in September, according to ESPN.
It appeared from video of the incident that Gallinari was hit in the face by Kok during a free-throw attempt. He shoved Kok and, after Kok took a swing in return, Gallinari landed a punch to the Dutch player's face.
According to USA Today, the Italian Federation said Gallinari suffered a fracture from the punch.
Gallinari signed a three-year, $65 million deal with the Clippers in July, moving from the Denver Nuggets in a sign-and-trade agreement.