Helmand governor Naeem Baloch told Karzai during a meeting in Kabul about the shooting, which comes as the US and Afghanistan wrangle over a deal to allow some US troops to remain in the country after this year.
The US-led NATO coalition in Afghanistan issued a statement expressing "deepest sympathies to the family who suffered the loss of a loved one" in the incident on Wednesday and vowing to investigate "what happened and why".
Karzai made a surprise decision not to sign the agreement promptly despite having vowed to do so, leading to the threat of a complete withdrawal of NATO troops by the end of 2014.
"We condemn the killing of this boy in the strongest terms," presidential spokesman Aimal Faizi told AFP.
More From This Section
"We have been calling for the complete end of military operations in residential areas. This demand has not been taken seriously by foreign troops and the result is civilian casualties including women and children."
Faizi said a ban on military operations in civilian areas was one of the Afghan conditions of signing the BSA.
"The ball is the US court," he said. "We are waiting for practical steps to be taken to end these operations and for the launch of a peace process. We believe the US can deliver on these demands."
Signing the BSA is a precondition for the delivery of billions of dollars in Western aid for Afghanistan, which will hold an election to choose Karzai's successor in April.
The BSA would see several thousand US troops remain in Afghanistan to provide training in the fight against Taliban insurgents after the NATO combat mission ends in December.