Ehsanullah Fazli, a doctor at the Asadabad hospital, said most of the wounded were children and some were in a critical condition.
"I was on my way to school. Their tyre burst, and then a soldier hurled a hand grenade from the convoy," Abdul Wahab, a 12-year-old boy, told the Reuters news agency as he lay in a hospital bed with shrapnel wounds in his legs.
Other victims at the Asadabad hospital gave similar accounts.
However, Sayed Fazelullah Wahidi, the provincial governor, said that US troops showed him fragments of a Russian-made grenade, an implication that someone in the crowd caused the blast.
At least three US troops were also reportedly wounded in the incident.
Abdul Jalal Jalal, Kunar's police chief, said that officials were investigating who threw the grenade into the crowd.
Civilian casualties
US and Nato officials have conceded that civilian casualties in Afghanistan must be reduced if they are to succeed against the Taliban.
Kabul has repeatedly complained that such incidents turn people against the Afghan government and the presence of foreign troops.
On Monday, the Pentagon acknowledged that procedures had been violated during an air raid in which the Afghan government says 140 civilians were killed.
Washington says up to 35 civilians were among the more than 80 people killed in western Farah province in early May.
In in the southern province of Zabul on Tuesday, police said that more than 40 Taliban fiighters had been killed in a week-long operation to stabilise the area before elections in August.
"During one week in different incidents and different areas, more than 40 Taliban fighters were killed and most of their bodies were left in the fighting area," Abdur Rahman Sarjang, the provincial police chief, said.
Six suspects were arrested and a haul of motorcycles and weapons were seized, he said.
Sarjang said the operation would continue until the area was stable enough to allow people to cast their ballots at the August 20 presidential and provincial council elections.