Slamming the US for the detentions and the arrest warrants, Erdoğan said, "What kind of a law is that?"
Mr Erdogan's security detail returned with him to Turkey after his visit, so it is unclear if any will face legal repercussions in the US.
Turkey's Ministry of Foreign Affairs said in a statement that it told the us ambassador that it believes the decision to issue arrest warrants is "wrong, biased and lacks legal basis".
Video footage showed men in suits charging past police to kick and punch protesters. He can be seen running past police officers and kicking at least two people who had fallen to the ground, including a man who had been kicked in the face.
Washington police, more accustomed to protecting visiting dignitaries from protesters than protecting protesters from official delegations, appeared to be caught off guard by the attack and struggled to contain the Turks. Eleven people were injured in what D.C. police described as a "brutal attack".
"Our understanding is those two individuals have Canadian addresses, but we're unable to confirm if they're in fact citizens", a police spokesperson said.
Sinan Narin of Virginia was arrested at his job near McLean, Virginia and charged with felony aggravated assault and misdemeanor assault or threatened assault in a menacing manner.
Eyup Yildirim was also arrested Wednesday, and his case is pending extradition from New Jersey.
When asked how DC authorities meant to apprehend the police and guards now in Turkey, Newsham said "the State Department is going to try to work that out".
He said many were being sought on assault charges, and other counts.
Narin and Yildirim were both participants in the protests, according to a us official familiar with the case. And previous year, the police and members of Mr Erdogan's security team clashed with demonstrators outside the Brookings Institution in Washington. "We make sure they are safe, but we also make sure they follow our laws", she said.
Bowser said the USA capital hosts millions of people "who come to the seat of their government to protest peacefully". Police are expected to release photos of the other 8 possible suspects and appeal to the public for information on their identities, the official said.
"The brawl in front of the Turkish ambassador's residence was caused by the failure of local security authorities to take necessary measures". The Turkish Foreign Ministry then summoned America's ambassador to address about the treatment of the detained security guards.
It is unlikely that Turkey would extradite the men to the United States. The Turkish undersecretary has told US ambassador John Bass that police in Washington did not take security precautions against the protesters, who Turkey says were associated with outlawed Kurdish militants.
"Any additional actions regarding execution of these warrants will be weighed by the State Department as appropriate under relevant laws and regulations", he added, noting the State Department has been helpful in supporting the investigation, and insisting no one from the Turkish embassy has been implicated.
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