Eight people were killed in the attack, which Syria says the U.S. perpetrated against civilians. The United States has yet to officially confirm its role in the raid, but U.S. officials have said on condition of anonymity that the raid killed a top operative of al-Qaida in Iraq, reported the Associated Press.
A demonstration is planned in Damascus on Thursday to protest the killings. A source familiar with the planning for the event said it would not be held near the embassy, reported Reuters.
"There are no plans to march on the embassy. An American flag may be set alight but the protest will be peaceful," the source said.
A statement on the U.S. Embassy's Web site warns that "the American community in Syria should be aware that unforeseen events or circumstances may occur that could cause the U.S. Embassy in Damascus to close to the public for an unspecified period of time."
The raid has further strained tense relations between Syria and the United States, which has accused Syria of failing to stop foreign fighter from passing over the border into Iraq to carry out attacks against U.S. forces.
The Syrian government on Tuesday said it would close an American school and cultural center in Damascus. However, both remained open on Wednesday.
Syria has also called on the U.N. Security Council to take action to prevent similar attacks in the future.
Deputy Foreign Minister Fayssal al-Mekdad said Syria is still waiting for an official explanation from Washington and Baghdad about "the unacceptable violation of Syrian sovereignty," reported Reuters.
Syria also indicated that the attack could weaken progress with Iraqi-Syrian relations by postponing a November meeting of the Syrian-Iraqi high commission.
Baghdad was slow to condemn the attack, but did on Tuesday and Iraqi Prime Minister Nuri al-Maliki said Iraq should not be used to stage attacks on other nations.
Iraq said Wednesday it will share with Syria the results of an investigation it has opened into the raid.
"The Syrian side will be provided with all details and information when investigation is concluded," the government's National Media Center said in a statement.