Cpl. Wassef Ali Hassoun is an American Muslim of Lebanese origin. A videotape aired Sunday showed him blindfolded, with a sword brandished over his head.
Hassoun's father pleaded Monday for his son's release saying the Marine was only doing his duty as a soldier.
Another Muslim Amjad Hafeez, a Pakistani driver, was kidnapped in Iraq last week.
The kidnappers said Hafeez was caught during an attack on the Iraqi city of Balad.
They want Pakistan to close its mission in Baghdad or they will behead Hafeez.
His mother has pleaded for his life.
The militants are also holding three Turkish Muslims. Two of them have been missing since June 1 but were confirmed as kidnapped Monday when the militants released their pictures.