At other sites, demonstrators burned tires or American flags. Afghan police and international troops fired guns in the air to disperse the crowds.
The protests sparked clashes with Afghan security forces that left at least five demonstrators dead. A Norwegian soldier was wounded by a hand grenade hurled into a coalition compound.
On Wednesday, six people died in protests in Kabul and three other provinces.
The civil unrest comes at a time when Afghan President Hamid Karzai is trying to negotiate a long-term partnership agreement with the United States to govern the activities of U.S. forces in Afghanistan after 2014, when most foreign combat troops will have left or taken on support roles.
Karzai called for calm until an investigation is completed, but the incident highlighted the fitful and often strained relationship of the two nations.
White House press secretary Jay Carney told reporters aboard Air Force One that Obama's apology to Karzai was "appropriate given the sensitivity" of the issue. He said the apology was part of a three-page letter to the Afghan leader. Presidential apologies are rare, but he noted that former White House press secretary Dana Perino apologized on behalf of President George W. Bush in 2008 after a U.S. serviceman shot a Quran.
Tommy Vietor, a spokesman for the National Security Council at the White House, said Obama's letter, which addressed issues being negotiated in the partnership document, was delivered by Ryan Crocker, the U.S. ambassador to Afghanistan. In the letter, Obama expressed "regret and apologies over the incident in which religious materials were unintentionally mishandled," Vietor said.
Karzai met Thursday with parliamentarians - many of whom had been particularly vitriolic Wednesday in calling for Afghans to wage a holy war against international forces. The Afghan president told the lawmakers they were right to raise their voices against the desecration of Islam's holy book, but said a government investigation was the appropriate way to handle the case, according to a statement issued by his office.
The statement said Karzai told the lawmakers that a U.S. officer responsible for the burning "didn't understand" what he was doing and that the United States had "accepted the mistake of its officer."
The coalition said the investigation is still under way.
The unrest started Tuesday when Afghan workers at the sprawling American base north of Kabul noticed that Qurans and other Islamic texts were in the trash that coalition troops dumped into a pit where garbage is burned. Some Afghan workers burned their fingers as they tried to salvage some of the books. Afghan government officials said initial reports indicated four Qurans were burned.
The materials had been taken from a library at Parwan Detention Facility, which adjoins the base, because they contained extremist messages or inscriptions. Writing inside a Quran is forbidden in the Islamic faith, and it is unclear whether the handwritten messages were found in the holy book or other reading materials.
A military official said it appeared that detainees at the prison were exchanging messages by making notations in the texts.
A delegation of Afghan religious leaders, lawmakers and government representatives visited Bagram as part of the investigation. They issued a statement late Thursday calling for an end to protests and accused insurgents of infiltrating the gatherings to foment violence. They said they expected those responsible for the Quran burning to be prosecuted through the U.S. military court system.
The Taliban used the opportunity to incite more attacks on foreign forces. In a statement, Taliban spokesman Zabiullah Mujahid described the burning as an "unforgivable crime." He urged Afghan army and police to become "real sons of the nation" by turning their guns on coalition forces.
Afghan authorities said demonstrations were staged inside the capital and in seven of Afghanistan's 34 provinces.
The deadliest was held outside an American base in the Khogyani district of Nangarhar province in eastern Afghanistan. Two protesters were killed by Afghan police and an Afghan soldier turned his gun on American troops, killing two.
It was the latest in a rising spate of incidents where Afghan soldiers or police or militants wearing their uniforms, have shot and killed U.S. and NATO service members.
District chief Mohammad Hassan said the two Americans were shot by a member of the Afghan National Army stationed at the base. Pentagon press secretary George Little later confirmed the two U.S. troops had died.
In northern Afghanistan, more than 10,000 people demonstrated at four locations in Baghlan, the capital of Baghlan province. Sayed Zamanuddin, deputy provincial police chief, said protesters tried to burn down a police office and then fired at police. He said the police returned fire. One person was killed.
In the south, two people were killed in Dihrawud district of Uruzgan province, said Fareed Ayal, a spokesman for the provincial police. He said insurgents infiltrated the demonstration and shooting broke out between the police and those in the crowd of about 2,000.
Also in the north, protesters hurled rocks and tried to remove the razor wire from the perimeter of a U.S. base in Mehterlam, the capital of Laghman province.
"The burning of Quran broke our hearts," protester Mohammad Issa said.
Police broke up the demonstration by using water cannon, batons and by firing above the heads of the demonstrators.
Hundreds of other Afghans protested peacefully on the eastern outskirts of Kabul, outside Bagram Air Field, in Jalalabad in the east, in Faryab province in the west and in Khoshi district of Logar province, where demonstrators burned a U.S. flag.
"Apologies are not enough," declared Mohammad Qasim Sediqi, leader of the Khoshi district council. The culprits "have to be put on trial and culprits should face the law."
Associated Press photographer Rahmat Gul in Mehterlam and AP writers Patrick Quinn and Heidi Vogt in Kabul, Mirwais Khan in Kandahar and Pauline Jelinek and Jim Kuhnhenn in Washington contributed to this report.