Coalition airstrikes on friendly targets have helped widen a rift between Afghan President Hamid Karzai and the US, cementing his resolve not to sign a security deal to let US troops remain in the country after 2014.
A spokesman for Mr Karzai said: "We condemn the attack on the Afghan National Army. The president has ordered an investigation."
The airstrike also seriously wounded at least eight other soldiers, according to district governor Khalilullah Kamal.
He said: "Right now a discussion in the province is going on between Afghan officials and foreign forces to find out the reason for this attack."
Coalition forces said the bombing was an accident.
They said in a statement: "We value the strong relationship with our Afghan partners and will determine what actions will be taken to ensure incidents like this do not happen again."
The Afghan Defence Ministry said the bodies of the dead and wounded personnel had been transferred to Kabul, adding a delegation had been sent to investigate.