Police said 11 people had been injured in the brawl, and two people, Ayten Necmi, 49, of NY and Jalal Kheirabaoi, 42, from Virginia had been arrested at the scene and charged with assault and assault on a police office respectively.
"We are communicating our concern to the Turkish government in the strongest possible terms", the State Department said in a statement.
Videos posted to social media showed a group of men in suits punching and kicking protesters, including a woman lying down, while police struggled to stop the violence. Two men were injured in the head while attendees were helping demonstrators in a state of confusion.
Mr Erdogan was in the U.S. capital on Tuesday to meet with President Donald Trump.
DC Mayor Muriel E. Bowser decried the incident as a "pretty savage beating" and said police and all city officials were determined to resolve the situation. So-called "diplomatic agents" and "administrative and technical staff" enjoy complete immunity from the criminal jurisdiction of US courts, according to State Department guidance for law enforcement.
In September 2011, a brawl erupted between members of then Prime Minister Erdogan's security detail and United Nations security officers in the halls of the United Nations General Assembly in NY.
It's not clear to what extent President Erdogan's full security team was involved. Turkish security officials were seen coming out of the ambassador's residence, then retreating to the residence grounds, a law enforcement official said, adding that the same scenario played out a couple of times.
Two people were arrested at the scene of the altercation, which erupted after Erdogan met with President Trump at the White House.
The Turkish Embassy said in a statement the protesters were affiliated with the outlawed PKK.
Germany and France agree to draw up roadmap on developing EU
Macron, in his third day in office, will host European Council President Donald Tusk at the Elysee Palace for a private dinner. Le Drian, also a strong supporter of Macron, had been considered for the prime minister's post, French media reported.
Washington police were initially overwhelmed but quickly drafted in reinforcements, quelling the fight and making two arrests, but not before 11 people including a United States officer were hurt.
Turkish Foreign Minister Mevlut Cavusoglu said on Thursday he wanted Brett McGurk, the U.S. special envoy to the coalition against Islamic State, removed from his post, saying he supported Kurdish militants.
This is not the first time Erdogan's security guards have faced scrutiny in the USA capital.
There may be issues with diplomatic immunity, Police Chief Peter Newsham said at a news conference.
The statement said a Turkish-American was "seriously injured" in self-defense.
He added several of Mr Erdogan's security personnel were armed with guns, which made the incident especially "dicey" for local police officers trying to keep protests peaceful. The group is considered a terrorist group by both Turkey and the United States.
Republican Senator John McCain also censured the "thuggish behavior".
"Gail told them, "He's a journalist and we invited him".