Erdogan, who met with US President Donald Trump at the White House just hours before the fighting broke out on Tuesday, briefly witnessed the brawl in which nine people were left requiring hospital treatment.
A video shows the Turkish President standing near his car as the fighting ensued, before turning his back and walking into the embassy.
On Wednesday the Turkish Ambassador to the US Serdar Kılıç was summoned to the State Department, a senior State Department official told CNN.
"The conduct of Turkish security personnel earlier this week is deeply disturbing," the official said. "The State Department has raised its concerns about these events at the highest levels and a thorough investigation that will allow us to hold the responsible individuals accountable is of the utmost importance to us."
The official also confirmed that two members of Erdogan's security detail "were briefly detained during the altercations and subsequently released" and returned to Turkey with Erdogan.
On Thursday, Arizona Sen. John McCain, the Republican chairman of the Senate Armed Services Committee, called for the Turkish ambassador to be kicked out of the US.
"We should throw the Turkish ambassador out of the country, we should identify those people that performed these unlawful acts of beating people up and they should be charged," McCain told reporters.
McCain and his Democratic colleague Sen. Dianne Feinstein of California also wrote a letter to Erdogan condemning the violence.
The State Department told CNN on Thursday that it is "working closely with the local authorities concerning this incident," but declined further comment as the investigation is ongoing.
The Turkish embassy claimed that the protesters were "affiliated with the PKK (Kurdistan Workers Party)" -- a banned separatist group in Turkey -- and had assembled without permission.
The protesters "began aggressively provoking Turkish-American citizens who had peacefully assembled to greet the President," the embassy said in a statement released late Wednesday.
"The Turkish-Americans responded in self-defense," the statement said. "We hope that, in the future, appropriate measures will be taken to ensure that similar provocative actions causing harm and violence do not occur."