Violent anti-American protests have erupted across Afghanistan since the U.S. military Tuesday apologized for what it said was the accidental "improper disposal" at Bagram Air Base of religious materials, including copies of the Muslim holy book.
The United States is working with the Afghan government to investigate the incident.
"We will take the appropriate steps to avoid any recurrence, including holding accountable those responsible," Obama said in the letter delivered by U.S. Ambassador to Afghanistan Ryan Crocker, Karzai's office said.
White House spokesman Jay Carney said the letter to Karzai was a follow-up to a telephone conversation between the two leaders.
"I think that the message that we're trying to convey here is that this was inadvertent. We take it very seriously," Carney told reporters aboard Air Force One Thursday.
Carney said NATO Commander Gen. John Allen, "is taking steps to ensure that this kind of thing can't happen again by instituting training on the handling of religious materials."
An Afghan soldier, apparently angry over the burning of Korans, fatally shot two U.S. troops and wounded four others, Afghan officials said.
Carney told reporters the expression of regret by Obama was part of a letter to Karzai "on a number of topics, including the Afghan-led reconciliation process and our bilateral relationship with Afghanistan, in which he included an expression of his regret and apologies for the inappropriate and inadvertent mishandling of religious materials."
Carney noted that Dana Perino, when she was press secretary for former President George W. Bush, "expressed apologies on behalf of the president" after a 2008 incident in which a U.S. serviceman "apparently shot or did damage to a Koran."
"And that's appropriate for the same reason, because our concern -- this president's concern, as was surely the case with President Bush, is the safety and security of our men and women in uniform, as well as our civilians in Afghanistan," Carney said.
"And one of the reasons that it's appropriate to express our sincere apologies for this incident is that the kind of reaction that it can cause risks putting our men and women in harm's way, or in further risk than they already are," he said. "So I think that precedent is a useful one to look at."
It remained unclear how many Korans were involved, a military official told CNN Thursday. The official said American troops at the base would have been unable to read the texts, which could have contributed to the error.
Another military official said earlier the materials had been removed from a detainee center's library because they carried "extremist inscriptions" on them and there was "an appearance that these documents were being used to facilitate extremist communications."
Violent protests took place Thursday at U.S. and NATO military facilities across Afghanistan, including one demonstration in which police apparently shot and killed two protesters.
The Taliban in Afghanistan called on Muslims to attack NATO facilities and convoys and kill military personnel, CNN reported.
In an e-mail message, the Taliban accused "the invading infidel authorities" of trying to calm the situation with two "so-called shows of apology, but in reality they let their inhuman soldiers insult our holy book."
The e-mail urged Afghans to seek revenge "until the doers of such inhumane actions are prosecuted and punished."
"We should attack their military bases, their military convoys, we should kill their soldiers, arrest their invading soldiers, beat them up and give a kind of lesson to them that they never dare to insult the holy Koran," the message said.
The International Security Assistance Force said in a statement two military personnel were killed in eastern Afghanistan Thursday by "an individual wearing an Afghan National Army uniform" but didn't identify the troops' nationality.
CBS News said an Afghan official said the dead and wounded in the attack in the eastern province of Ningarhar were American. The official said the shooting seemed to be motivated by the burning of Korans at the Bagram Air Base north of Kabul, but did not elaborate.
Celebrity Families of 2014 [PHOTOS]