, posted to YouTube, shows the soldiers as they talk into a camera, standing in front of a pair of U.S. and Qatari flags.
The soldier on the left laughs as he says in a slightly slurred voice that he's speaking from "an unknown disclosed location in a place called ..." -- then makes an indistinct sound to further indicate his location is secret.
He gestures at the Qatari flag behind him in an exaggerated fashion, while the second soldier is laughing.
The second soldier regains her composure and tells her colleague they "need to be serious," before breaking into laughter again. That prompts a voice off camera to chime in, "This is as serious as it gets."
While the video appears lighthearted, the Qatari government was not amused. They took it seriously enough to summon the American ambassador, Dana Shell Smith, so she could explain the video.
In Qatar, it is illegal to insult or express a lack of respect for the flag and violators face prison terms of up to three years, and a potential 200,000 riyal fine (about $55,000).
Smith was quick to respond on Twitter, writing in Arabic, "Dear followers, please trust that those soldiers were making fun of themselves and not the country of Qatar."
"I offered my apologies to the Qatari government and I asked American military leaders to investigate the incident and take the proper ethical measures," she added.
At the State Department briefing Thursday, Deputy Spokesman Mark Toner told reporters the situation was "an example of the ability of social media to send the wrong message."
"Apparently that video offended some who saw it -- some Qatari citizens who saw it," said Toner, who explained that the video was made at an Army social event. "Amb. Smith took it upon herself as ambassador to Qatar to issue an apology on Twitter and also convey in person her apology to the Qatari Ministry of Foreign Affairs."
Pentagon Spokesman Peter Cook offered an apology as well, saying, "We sincerely regret any offense the video may have caused in Qatar."
Cook also insisted that members of the military are held to the "highest standards of personal conduct," and promised the Pentagon would take "appropriate action."