The Syrian Foreign Ministry condemned the action as a violation of international diplomatic conventions in a statement carried by Syrian state television.
Moscow called it "worrying and disappointing".
Worsening relations between the United States and Russia over Ukraine have clouded efforts by the two countries to broker a peace deal in Syria, which is in its fourth year of civil war.
The Russian Foreign Ministry suggested Washington's actions were aimed at "regime change" to oust Syrian President Bashar al-Assad whom Russia has protected with diplomatic support at the U.N. Security Council and with arms deals.
"The aim of regime change in Damascus prevails over the task of disarming Syria of its chemical weapons and helping millions of Syrians who have suffered from the armed conflict," the ministry said on its website.
The Syrian Foreign Ministry said: "The United States of America committed a clear violation of the Vienna conventions on diplomatic relations and consular relations by resorting to an arbitrary measure."
It said the United States had "torpedoed the fundamental legal principle of consular work by involving honorary consulates in political matters".
In Washington, State Department spokeswoman Jen Psaki rejected Syria's charge, saying: "We are abiding by not only the law but every aspect of the Vienna Convention on diplomatic relations."
She said Syria could appoint a "third party state" or a member of its US-based staff, who is either a US citizen or US permanent resident, to care for the embassy premises and archives. The decision would need the approval of the State Department, Psaki added.
The State Department has given Syrian diplomats until the end of March to leave the country, while administrative staff have until April 30 to pack up.
Syria announced on March 10 it would stop providing consular services in the United States. The State Department said on Tuesday the missions' closure did not constitute a break in diplomatic ties with Syria despite failed peace initiatives to end the conflict in the Arab country.