Karzai's brothers were unharmed in the brief battle, which began during meetings with local people at a mosque near Najiban and Alekozai villages, but a soldier was killed and a civilian wounded. The area is a Taliban stronghold and a supply route.
The Taliban had earlier threatened reprisals for the weekend shooting spree, which came weeks after deadly riots across the country over the burning of copies of the Koran by U.S. troops at NATO's main base in the country. That violence led to calls to accelerate a 2014 goal for the exit of most foreign combat troops.
"The Islamic Emirate once again warns the American animals that the mujahideen will avenge them, and with the help of Allah will kill and behead your sadistic murderous soldiers," Taliban spokesman Zabihullah Mujahid said in a statement, using the term by which the Islamist group describes itself.
The grim warning, which was unlikely to have any impact on heavily-protected NATO soldiers on the ground, followed the February beheading of four Afghan men by insurgents in a country where such killings are relatively rare.
Tuesday's attack, which was carried out despite tight security around Karzai's siblings, Kandahar governor Tooryalai Wesa and Tribal Affairs Minister Asadullah Khalid, underscored the insurgents' ability to strike at fledgling Afghan government forces.The first protests over Sunday's massacre also broke out in eastern city Jalalabad, where around 2,000 demonstrators chanted "Death to America" and demanded President Karzai reject a planned strategic pact with Washington that would allow U.S. advisers and possibly special forces to remain in Afghanistan beyond 2014.
An unnamed U.S. soldier -- reported to have only recently arrived in the country -- is accused of walking off his base in Kandahar province in the middle of the night and gunning down at least 16 villagers, mostly women and children.
A U.S. official said the accused soldier had suffered a traumatic brain injury while on a previous deployment in Iraq.
U.S. President Barack Obama, speaking after a phone call with Karzai -- who is said to be furious over the latest deaths -- said the shootings had only increased his determination to get American troops out of Afghanistan as planned.
However, Obama cautioned there should not be a "rush to the exits" for U.S. forces who have been fighting in Afghanistan since late 2001 and that the drawdown set for the end of 2014 should be done in a responsible way.
The soldier, from a conventional unit, was based at a joint U.S.-Afghan base used by elite U.S. troops under a so-called village support program hailed by NATO as a possible model for U.S. involvement in the country after the 2014 drawdown.
Such bases provide support to local Afghan security units and provide a source of security advice and training, as well as anti-insurgent backup and intelligence.
"CAN NO LONGER BE CALLED ROGUE"
A spokesman for the Kandahar governor Wesa said tribal elders in the area of the massacre would urge against protests and work to dampen public anger if the investigation process was transparent."They are supporting the government and will accept any conclusion by the investigators. Today we have meetings with people in the area and all will become clear," spokesman Ahmad Jawid Faisal said.
NATO officials said it was too early to tell if the U.S. soldier would be tried in the United States or Afghanistan if investigators were to find enough evidence to charge him, but he would be under U.S. laws and procedures under an agreement between U.S. and Afghan officials.
Typically, once the initial investigation is completed, prosecutors decide if they have enough evidence to file charges and then could move to an Article 32 or court martial hearing.
NATO's top commander in Afghanistan, Marine General John Allen, has promised a rapid investigation of the massacre, while security was being reviewed at NATO bases across the country.
U.S. Defense Secretary Leon Panetta said on Monday that the death penalty could be sought in the U.S. military justice system against the soldier, but portrayed the shooting as an isolated event that would not alter withdrawal plans.
While Afghan MPs in parliament called for a trial under Afghan law, Karzai's office was understood to accept that a trial in a U.S. court would be acceptable provided the process was transparent and open to media.
Afghan MPs protested against the shooting for a second day in the capital Kabul by walking out of a session.
Analysts said the incident would complicate U.S. efforts to reach agreement with the Afghan government on a post-2014 security pact before a May summit in the U.S. city of Chicago on the future size and funding of Afghan security forces.
Thomas Ruttig of the Afghanistan Analysts Network said that despite NATO and White House references to the killings as the work of a "rogue" soldier, similar events had happened before, including a "kill team" apprehended in Kandahar in 2010.
"In the stress of an environment of escalated violence - by both sides, but particularly after Obama's troop surge in early 2009, it looks as if most soldiers simply see Afghanistan as a whole as 'enemy territory' and every Afghan as a potential terrorist. This can no longer be called 'rogue'," Ruttig said.