Abushagur said Thursday the suspect who is in custody, along with "three or four" others being sought, are Libyans, CNN reported.
He said there are "suspicions that these people belong to some extremist group, but ... that is something we don't know."
"This is not acceptable to the Libyan people; this is not acceptable to our values," Abushagur said of the Tuesday attack in which U.S. Ambassador to Libya Christopher Stevens and three other U.S. diplomats were killed. "We are taking this very, very seriously."
Protesters marched in Tehran Thursday and the U.S. Embassy in Kuwait was closed amid growing turmoil over a movie produced in the United States disparaging the Prophet Muhammad. Protests were also carried out in front of U.S. embassies in Tunisia, Morocco, Israel and Sudan, CNN said.
In Sanaa, Yemen, four protesters were killed in clashes with security forces as hundreds stormed the U.S. Embassy Thursday, CNN reported. Protesters tried to storm the embassy compound in Cairo Thursday.
Hundreds of protesters, mostly students, gathered at the Swiss Embassy in Tehran, which serves as host to the U.S. Interests Section in Iran, Mehr News Agency reported.
"Based on Article 7 of the U.N. Charter, any country which endangers the global security should account for its acts and we urge international organizations to seriously ask the U.S. to present explanations," the protesters said in a statement condemning the anti-Islam film.
They urged Muslims to condemn the movie and boycott YouTube, where a clip from the movie was posted.
Some of the protesters in Tehran burned a U.S. flag and shouted "Death to the U.S." and "Death to Israel."
The U.S. Embassy in Kuwait was being closed ahead of Friday's customary prayers, Press TV reported.
White House press secretary Jay Carney told reporters Thursday the U.S. government "had nothing to do with" the movie at the center of the spreading protests.
"We reject its message and its contents," Carney said. "We find it disgusting and reprehensible. America has a history of religious tolerance and respect for religious beliefs that goes back to our nation's founding. We are stronger because we are home to people of all religions, including millions of Muslims.
"We also believe there is no justification at all for responding to this movie with violence," Carney said. "Islam respects the fundamental dignity of human beings and it violates that dignity to wage attacks on innocents."
Witnesses in Sanaa Thursday said protesters broke through an outer perimeter and set fire to a building inside the compound before being forced to retreat, The New York Times reported.
Protesters burned a U.S. flag and set fire to tires and two vehicles. Security officials fired into the air, The Wall Street Journal said. Yemeni officials said a number of protesters were wounded and there were some arrests.
President Abdu Rabbo Mansour Hadi of Yemen said in a statement he "extends his sincere apologies to President Obama and to the people of the United States of America" for the attack.
The embassy, on its website, had warned of the potential of protests spreading to Yemen.
The White House said President Barack Obama called Hadi Thursday to discuss the assault on the embassy in Sanaa and "express concern about the security of American personnel and diplomatic facilities in Yemen."
Obama thanked Hadi for condemning Thursday's violence and "expressed appreciation for the cooperation we have received from the Yemeni government and underscored the importance of working together to ensure the security of U.S. personnel going forward," the White House said in a statement.
The statement said Hadi "committed to doing everything possible to protect American citizens in Yemen, and said he had deployed additional security forces around the U.S. Embassy," and Obama once again condemned any effort to denigrate Islam.
Earlier Thursday in Cairo, riot police fired warning shots and lobbed tear gas at hundreds of protesters outside the U.S. Embassy, CNN reported. Similar demonstrations against the movie, of which a 13-minute trailer was posted on YouTube, occurred outside the mission Tuesday and Wednesday.
Six police officers sustained minor injuries in the latest protests, Egyptian Interior Ministry spokesman Alla Mahmoud said. Some protesters were treated on the scene for injuries.
Witnesses said the demonstrators threw Molotov cocktails and rocks, CNN reported.
A U.S. Navy destroyer took up a position off Libya's coast and another was en route, and about 50 elite Marines deployed to Tripoli were in place Thursday to provide security at the U.S. Embassy, officials said.
Agents of the CIA, the FBI and other agencies were marshaled to identify and pursue the attackers who assaulted the U.S. Consulate in Benghazi Tuesday, killing the four Americans in an intense, 4-hour firefight for control of the mission.
U.S., Libyan and European officials have suggested militants used the demonstration as cover to stage a prearranged attack on the 11th anniversary of the Sept. 11 terror attacks on New York and Washington.
Obama spoke with Libyan President Mohamed Magariaf Wednesday, thanking him for his condolences and expressing appreciation for the government's assistance in responding "to this outrageous attack," a readout of the conversation provided by the White House indicated.
In a conversation with Egyptian President Mohammed Morsi Thursday, Obama "underscored the importance of Egypt following through on its commitment to cooperate with the United States in securing U.S. diplomatic facilities and personnel," a readout indicated.
Stevens' body was not located until dawn Wednesday, when he was found dead at a Benghazi hospital, U.S. and Libyan officials told The New York Times. It was the first time since 1979 a U.S. ambassador had been killed in a violent assault.
Officials told several news organizations the attackers were organized, well trained and heavily armed, and they appeared to have some level of advance planning.