The Dodo did a little research, as Smith urges. There is a whole manual issued by the U.S. military that gives rough guidelines on dealing with stray animals, and recognizes that in some areas units will decide to put in place a policy to euthanize stray animals. "Euthanizing a stray animal can, on occasion, be considered a humane action. It is always the last resort and should be practiced only when rescue and adoption are not possible," the report says. The guide encourages military units to educate the public about this to keep up public perceptions of the military: "Provide information about the hazards caused by stray animals and stress the link to personnel health, safety and welfare and protection of wildlife from these animals."
While the military does sometimes euthanize stray animals, it says nothing that condones calling the animals assholes or posting the grim photo on Facebook and bragging about it.
One animal advocate posted to U.S. Army Fort Lee's Facebook page, asking the question, "Is there any comment about what our soldier, Chris Smith, who defends our country, AMERICA, did with the killing of those dogs? I think AMERICA would like to know what you are doing about it. Court Marshall is in order."