Jaminet, who plays for Top 14 club Toulon, was seen saying, "the first Arab I come across, I’ll throw him a headbutt," in a video that was widely shared recently.
The French team are currently playing in a summer tour in Argentina.
"FFR strongly condemns the remarks made by Melvyn Jaminet in a video recently posted on social networks. Such comments are totally unacceptable and contrary to the fundamental values of our sport," the FFR said in a statement.
"As a result, Melvyn Jaminet has been suspended with immediate effect and is leaving the France squad currently in Argentina."
Jaminet apologised for his comments in a post on Instagram.
"I understand that this may have hurt and offended many people, and I want to make it clear that these comments in no way reflect my values or those of the French rugby team," he wrote.
"Racism, in all its forms, is unacceptable and goes against everything I believe in."
The FFR added that an internal investigation is underway.
The decision was welcomed by French Minister of Sports Amelie Oudea-Castera, who called for zero tolerance against racism in a post on X.