Taliban militants opened fire Tuesday on an Afghan government delegation visiting one of the two villages in southern Afghanistan where a U.S. soldier is suspected of killing 16 civilians.

The gunfire killed an Afghan soldier who was providing security for the delegation, said Gen. Abdul Razaq, the police chief for Kandahar province where the visit took place. Another Afghan soldier and a military prosecutor were wounded in the attack, he said.

The attack in Balandi village came as the Taliban vowed to kill and behead those responsible for the 16 Afghan civilians killed Sunday.

The delegation, which included two of President Hamid Karzai's brothers and other senior officials, was holding a memorial service in a mosque for victims when the shooting started.

One of the president's brothers, Qayum Karzai, said the attack didn't seem serious to him.

"We were giving them our condolences, then we heard two very, very light shots," said Karzai. "Then we assumed that it was the national army that started to fire in the air."

He said the members of the delegation, which also included Kandahar governor Tooryalai Wesa and Minister of Border and Tribal Affairs Asadullah Khalid, were safe and headed back to Kandahar city.

Also Tuesday, hundreds of students in eastern Afghanistan held the first significant protest against the killings.

The students shouted angry slogans against the soldier in the eastern city of Jalalabad and carried banners calling for his public trial.

Some of the protesters were constructing an effigy of President Barack Obama that they planned to burn.

Sunday's killings in Kandahar province have caused outrage in Afghanistan, but have not sparked the kind of violent protests seen last month after American soldiers burned Muslim holy books.

The more muted response could be a result of Afghans being used to dealing with civilian casualties over a decade of war.

More details emerged Monday about the suspect, an Army staff sergeant who is currently detained in Afghanistan.

The suspect, whose name has not yet been released, is described to Fox News by military sources as an infantry sniper out of Joint Base Lewis-McChord in Washington state. He suffered traumatic brain injury in 2010 after a vehicle rollover accident in Iraq, though he later was deemed "fit for duty," a senior U.S. official said.

Sources also said the suspect had "family problems," possibly related to trouble in his marriage, before deploying for the fourth time.

The suspect will likely will stand trial in the U.S., military sources told Fox News -- and Defense Secretary Leon Panetta said he could face the death penalty.

Panetta said he may face capital charges, and that the U.S. must resist pressure from Washington and Kabul to change course in Afghanistan because of anti-American outrage over the shooting.

"We seem to get tested almost every other day with challenges that test our leadership and our commitment to the mission that we're involved in," Panetta told reporters traveling with him to Krygzystan. "War is hell."

Obama also expressed his shock and sadness and extended his condolences to the families of the victims, but said the horrific episode would not speed up plans to pull out foreign forces.

Sunday's attack in southern Kandahar province unfolded in two villages near a U.S. base. Villager Mohammad Zahir recounted to the Associated Press how an American soldier burst into his home in the middle of the night, searched the rooms, then dropped to a knee and shot his father in the thigh as he emerged from a bedroom.

"He was not holding anything -- not even a cup of tea," Zahir said.

The AP reported the suspect is a 38-year-old father of two. Fox News sources said he didn't turn himself in, but rather returned to the base and was detained. He was the only person thought to have left the base.

The shootings come as anti-Americanism already is boiling over in Afghanistan after U.S. troops burned Korans last month and a video of Marines urinating on alleged Taliban corpses was posted on the Internet in January.

An enraged Afghan President Hamid Karzai called it "an assassination, an intentional killing of innocent civilians" that cannot be forgiven. He demanded an explanation from Washington for the deaths, which included nine children and three women.

NATO and member countries said the slayings were a blow to the alliance's efforts to cultivate trust but would not affect the timeline to hand over security operations to Afghans by the end of 2014. The White House said U.S. objectives will not change because of the killings.

The soldier has been in the military for 11 years and served three tours in Iraq. His name was not released because it would be "inappropriate" to do so before charges are filed, said Pentagon spokesman George Little.

But Panetta, his first public remarks on the incident, said Monday evening the death penalty is a consideration as the military moves to investigate and possibly put the suspect on trial.

The soldier was deployed to Afghanistan on Dec. 3 with the 2nd Battalion, 3rd Infantry Regiment of the 3rd Stryker Brigade, 2nd Infantry Division, based at Joint Base Lewis-McChord located south of Seattle, a congressional source told the Associated Press, speaking on the condition of anonymity because of the sensitivity of the matter.

He was attached Feb. 1 to the village stability program in Belambai, a half-mile from one of the villages where the attack took place, the congressional source said.

Zahir told how he watched the soldier enter his house and move through it methodically, checking each room.

"I heard a gunshot. When I came out of my room, somebody entered our house. He was in a NATO forces uniform. I didn't see his face because it was dark," he said.

Zahir, 26, said he quickly went to a part of the house where animals are penned.

"After that, I saw him moving to different areas of the house -- like he was searching," he said.

His father, unarmed, then took a few steps out of his bedroom, Zahir recalled. Then the soldier fired.

"I love my father, but I was sure that if I came out he would shoot me too. So I waited." Zahir said. His mother started pulling his father into the room, and he helped cover his father's bullet wound with a cloth. Zahir's father survived.

After the gunman left, Zahir said he heard more gunshots near the house, and he stayed in hiding for a few minutes to make sure he was gone.

Fox News' Jennifer Griffin and Justin Fishel and The Associated Press contributed to this report.