Twelve people were injured after protests outside of the Turkish ambassador's residence, DC Police Chief Peter Newsham announced in a press conference.
It added that the United States is "communicating our concern with the Turkish government in the strongest possible terms".
A video posted online showed men in dark suits chasing protesters and punching and kicking them as baton-wielding police tried to intervene.
Police managed to separate the two groups by the time Erdogan arrived, but tensions exploded when a man breached the police line and ran across the street to attack a protester.
"The actions seen outside the Turkish Embassy yesterday in Washington, D.C. stand in contrast to the First Amendment rights and principles we work tirelessly to protect each and every day", the police statement said.
He says there must be a diplomatic response to the incident from the Trump administration.
He urged the Secretary of State and the Attorney General to immediately deal with this issue and bring all appropriate criminal charges before these individuals leave the US.
Arthur told CNN that some of the men who attacked them were part of the Turkish security detail.More news: Russian Federation making progress but much work ahead, says WADA
Despite Trump's greeting to the Turkish leader, the relationship has been strained by the United States' refusal to extradite a Turkish cleric living in Pennsylvania, whom Erdogan blames for orchestrating a July coup attempt against him.
The situation escalated after Erdogan's guards and other Turkish security personnel interfered and violently beat the protesters. Turkey is a longstanding North Atlantic Treaty Organisation ally but relations with the USA have been strained over the years since Erdogan's Islamist-leaning government took power, pursuing foreign and domestic policies often troubling to Washington.
The Turkish embassy said the protesters were linked to the Kurdistan Workers Party (PKK) and the demonstration was "unpermitted" and "provocative".
"We said we would not be in such an operation with you where you ally with terror organizations and so we said good luck", Erdogan said. One of those injured was a police officer, while all nine people hospitalized have been released.
Turkey has demanded today the removal of USA representative at the Global Coalition against the Islamic State (IS), Brett McGurk, for his support to the Kurdish militias established in northern Syria.
The city of Washington on Wednesday condemned a "brutal attack on peaceful protesters" by Recep Tayyip Erdogan's bodyguards.
At their meeting Tuesday, Trump said the USA would re-establish its military and economic partnership with Turkey, committing to backing Turkey's defense against both Islamic State and the PKK.
"Replacing McGurk would be beneficial", Cavusoglu told private broadcaster NTV in an interview on Thursday, after just returning from a visit with President Erdogan to Washington where they met President Donald Trump on Tuesday. "The American police let them attack us".