The two young boys did not appear to understand the questions, which were in English, but smiled at the camera and at times flashed the "thumbs up" gestures during the 30-second clip.
Spc. Robert A. Rodriguez, who is based at Fort Wainwright in Fairbanks, was ordered to remove the video from his site, Army spokesman Maj. Bill Coppernoll said Monday.
"The incident is currently under investigation, and the Army will take appropriate action based on the findings of the investigation," he said.
It wasn't immediately clear if Rodriguez shot the video or just posted it, and discovering that will be part of the Army's investigation, Coppernoll said.
The video is "disgraceful and clearly inconsistent" with standards expected of every soldier, he said.
Raleigh, North Carolina, television station WRAL first reported the video after another soldier stationed at Fort Wainwright shared it with friends in North Carolina, who took their concerns to the station.
WRAL aired part of the video and quoted from Rodriquez's Facebook page before the site was made private.
Above the Facebook video posting, which was titled, "future gay terrorist!," is written, "i got bored in iraq ... so I kept myself entertained!"
The boys are shown on a dirt road, facing a camera.
A voice is heard asking the boys, "Are you going to grow up to be a terrorist?"
When the boys show two thumbs up, the voice on the video says, "Yeah. All right. Cool. Yeah, terrorist."
There was no phone listing for Rodriguez in the Fairbanks area. Coppernoll said he did not know the soldier's hometown, but the video of the Facebook page shown on WRAL indicated Rodriguez listed Miami.
"For anybody to be so cruel and disrespectful to children of any country but especially a country that we are occupying is really disgraceful and repugnant," said Tim Stallard, a spokesman for Alaskans Together for Equality.
The anti-gay bias is disturbing, also, he said.
"It's ugly, as well, and it's sorry to see anybody expressing such unfortunate and strong bias against gay people," he said.