An Afghan man and his son take part in a protest in Kabul March 6, 2011. Hundreds of people, chanting "Death to America!",
At a meeting with his security advisers at which Petraeus was present, Karzai said civilian casualties by foreign troops were "no longer acceptable" to the Afghan government or to the Afghan people, Karzai's palace said in a statement.
Civilian casualties caused by NATO-led and Afghan forces hunting insurgents have again become a major source of friction between Karzai and his Western backers.
In the meeting, Petraeus apologized for the deaths of the nine children in eastern Kunar province last Tuesday, saying the killings were a "great mistake" and there would be no repeat.
"In return, the president said the apology was not enough and stressed that civilian casualties caused during operations by coalition forces were the main cause of strained relations between the United States and Afghanistan," the palace said.
"The people of Afghanistan are fed up with such horrific incidents and apologies or condemnation is not going to heal their wounds," it quoted Karzai as saying.
Hours before Karzai's statement, hundreds of people chanting "Death to America" protested in the Afghan capital against the recent spate of civilian deaths, in a sign of the simmering anti-Western feeling among many ordinary Afghans.
International concern over civilian casualties has grown, and the fallout from the recent incidents is even threatening to hamper peace and reconciliation efforts, with a gradual drawdown of the 150,000 foreign troops in Afghanistan to begin in July.
"DEEP REGRET"
Last Tuesday, two attack helicopters gunned down nine Afghan boys as they collected firewood in Kunar after a nearby foreign base had come under insurgent attack.
The incident, in a volatile area that has seen a recent spike in foreign military operations, prompted rare public apologies from Petraeus and his deputy.
President Barack Obama also expressed "deep regret" over the killings and the United Nations called for a review of air strikes.
There have been at least four incidents of civilian casualties by foreign troops in the east in the past two weeks in which Afghan officials say more than 80 people died.
Demonstrators marched through the center of Kabul, some carrying banners bearing pictures of blood-covered dead children they said were killed in air strikes by foreign forces.
"We will never forgive the blood shed by our innocent Afghans who were killed by NATO forces," said one protester Ahmad Baseer, a university student.
"The Kunar incident is not the first and it will not be the last time civilian casualties are caused by foreign troops."
Dozens of women were also among the protesters, a rare occurrence in a country where women are largely banned from public life. Using loudspeakers, some of the women chanted: "We don't want Americans, we don't want the Taliban, we want peace."
PROTESTERS BLAME BOTH SIDES
U.S. and NATO commanders have tightened procedures for using air strikes in recent years, but mistaken killings of innocent Afghans still happen, especially with U.S. and NATO forces stepping up operations in the past few months.
Although civilian casualties caused by foreign forces have decreased over the past two years -- mainly due to a fall in air strikes -- aid groups last November warned a recent rise in the use of air power risked reversing those gains.
Civilian casualties in Afghanistan rose 20 percent in the first 10 months of 2010 compared with 2009, according to U.N. figures, with insurgents responsible for more than three-quarters of those killed or wounded.
In the latest attack by insurgents, 12 civilians were killed on Sunday when their vehicle was hit by a roadside bomb in southeastern Paktika province, governor Mohebullah Sameem said.
But while insurgents are responsible for the large majority of civilian deaths, it is those by foreign forces which rile Afghans most. Many Afghans say militant attacks would not happen if international troops were not in Afghanistan.
"Killing civilians, whether it is the Taliban or foreign forces, is a crime," said protester Shahla Noori.
"Both the Taliban and Americans are responsible for the killings of thousands of civilians," she said.
(Additional reporting by Matt Robinson in Kabul and Elyas Wahdat in Khost; Writing by Jonathon Burch; Editing by Elizabeth Fullerton)