But the police said they are also working with the US State Department and Secret Service "to identify and hold all suspects" - suggesting that they are seeking to interview Erdogan's security detail.
Attacking the small group of protesters with their fists and feet, men in dark suits and others were recorded repeatedly kicking one woman as she lay curled on a sidewalk.
The protest turned violent, and nine people were injured.
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Video appears to show Erdogan's bodyguards violently breaking up a protest this week while Erdogan was visiting Washington.
In a statement, the Turkish Embassy blamed the violence on the demonstrators, saying they were "aggressively provoking Turkish-American citizens who had peacefully assembled to greet the president". "They were the ones who wanted to come and we helped them".
The State Department did not immediately respond to an AFP inquiry about the incident and Secretary of State Rex Tillerson ignored a question about it at a media photo opportunity. The department says the ambassador met Wednesday with Tom Shannon, the acting deputy secretary of state and highest-ranking career USA diplomat.
A congressional aide told The Associated Press that two members of Erdogan's security detail were detained on the scene Tuesday by diplomatic security agents. The statement said that the USA communicated its concerns to the Turkish government "in the strongest possible terms". The document says he struck an officer in the face while the officer was trying to separate protesters and that he "engaged in a physical confrontation" with the officer.
The biggest dispute between Turkey and the US recently has been the Trump administration's plans to arm Kurdish Syrian militants fighting the Islamic State group.
Later, speaking to reporters at the Turkish embassy, Erdogan underlined Turkey's concerns about the YPG, which Ankara regards as an extension of the Kurdistan Workers Party (PKK) militant group fighting a decades-old insurgency in southeast Turkey. Washington sees the YPG as distinct from the PKK and as a valuable partner in the fight against Islamic State in Syria. By facilitating Armenian Genocide denial, by remaining silent regarding Turkey's illegal blockade of Armenia, by disregarding its support for ISIS and al Qaida, by overlooking Turkey's suppression of its religious and national minorities and its widening repression against its citizens-at-large, the US has only emboldened the Erdogan regime to become more aggressive, both inside and outside of Turkey. Such groups will "have no safe quarter", Trump said.
Erdogan said he warned Trump that Turkey would combat YPG if the group posed any security threat.