His statement published on the government's Facebook page on Monday came a day after the Samoa Observer newspaper made public for the first time the existence and content of a seven-second clip that the outlet said was filmed at a Schmidt family reunion in Samoa in July 2007.
In a screenshot from the video, Schmidt holds a microphone in one hand with the other arm lifted, surrounded by others making the same gesture.
The newspaper reported Schmidt was heard saying into the microphone, "Raise your right hand, Heil Hitler!"
Schmidt, who was the deputy speaker of Samoa's parliament in 2007, did not dispute the report.
He has led the South Pacific island nation of 220,000 people since winning an August 2025 election.
"I recognise that those words and gesture were inappropriate and capable of causing offence and hurt and I apologise for using them," he said on Monday.
He said the video was filmed "at a private family gathering - not intended for the public".
He rejected "Nazism, anti-Semitism, racism and all forms of political extremism", the statement said.
What happened immediately before and after the moment depicted was unknown, the newspaper said, and Schmidt's statement did not explain further.
The episode unfolded as Pacific media groups decried Schmidt's ongoing rancour against the Observer, which has been barred from covering his and other government news conferences and events since November, when Schmidt accused the newspaper of misleading coverage.
On Friday, Schmidt confirmed claims that an Observer reporter's photos of him, taken in front of government buildings during an official event, were deleted from their camera by an aide.
He defended the move, saying the Observer, Samoa's largest newspaper group, was flouting its prohibition.
The Observer has called his ban "authoritarian".
Pacific media groups have called on the prime minister for a reversal, with the Journalists Association of Samoa saying it was "deeply disturbed" by the photo deletion incident.
