A ministry statement said that the USA strike wounded 25 Syrian pro-regime fighters it said were searching for militants near an oil refinery.
Syrian state-run media says the USA strikes killed and wounded dozens of tribal fighters, calling it a new effort "to support terrorism". The US-led coalition backs a Kurdish-dominated ground force in eastern Syria, which has an increasingly uneasy and sometimes antagonistic relationship with Mr Assad, who is also backed by Iran.
Regime forces were likely attempting to take an oilfield near Khusham.
Last year, US aircraft shot down a Syrian jet after pro-government forces attacked USA allied units in northern Syria.
"It was self-defense. Obviously, we are not getting engaged in the Syrian civil war", Mattis said.More news: Bristol-Myers Squibb shares jump on lung cancer breakthrough
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White added that all pro-government vehicles and personnel who turned around and headed west were not targeted during the coalition strikes. The Russians assured the Americans the US -backed forces would not be harmed.
The river has always been a dividing line between USA -backed forces and Russian- or Iranian-backed fighters of the Assad regime. A total of 25 pro-government fighters were wounded.
The Syrian Observatory said they were local tribesmen loyal to President Bashar al-Assad and Afghan Shia militiamen.
The SDF is led and mostly comprised of fighters from the People's Protection Units (YPG), the armed wing of the Kurdish Democratic Union Party (PYD) that controls vast swaths of Northern Syrian.
Following the incident, the Russian command at the air base of Hmeymim established channels of interaction with representatives of the worldwide coalition headquarters, which stated that the local oil refinery was controlled by the SDF and US military personnel, the statement said.
Recent developments indicate that the United States might be effectively shifting towards partitioning Syria altogether, Russia's Foreign Minister Sergey Lavrov warned earlier on Wednesday.