Russia has denied links to the aircraft, calling the claim “stupidity.”
Army Gen. Stephen Townsend said the move threatens a major escalation in the civil war.
Pompeo called to "reiterate U.S. opposition to the continued level of weapons and munitions being brought into the country."
The United States on Thursday accused Moscow of worsening the Libya conflict.
Libya has been in turmoil since 2011.
French President Emmanuel Macron on Wednesday accused Turkey of breaching an agreement to halt foreign interference in Libya.
Libya has been divided between two competing governments since 2015.
The Tripoli-based government of Libyan Prime Minister Fayez Sarraj has faced an offensive by the rival regime in the east.
A senior official in the LNA’s general command said they mistook the U.S. drone for a Turkish-made drone .
The announcement comes after two unmanned drone aircraft — one American — were lost near the capital last week.