It was unclear if they were members of Mr Erdogan's security or protesters.
Washington's police chief, mayor and several USA lawmakers condemned the violence.
Parts of the clash were caught on video by VOA's Turkish Service.
Metropolitan Police Chief Peter Newsham called Tuesday's attack on Washington's Embassy Row "brutal". Video shows people pushing past police to confront a small group of protesters across the street in Sheridan Circle.
Photos and videos posted on social media by witnesses showed a chaotic scene of flying fists, feet and police batons.
"What we saw yesterday - a violent attack on a peaceful demonstration - is an affront to DC values and our rights as Americans", D.C. Mayor Muriel Bowser said. Nor did they answer whether or not the Turkish personnel involved in the attack would be disciplined. "Someone was beating me in the head nonstop", she said.
The Turkish Embassy in a statement blamed the violence on demonstrators, stating they were "aggressively provoking Turkish-American citizens who had peacefully assembled to greet the president".
Two former senior administration officials said it was retired Army Gen. Michael Flynn, Trump's former national security adviser, who rejected such a move, telling the Obama team that the Trump team first wanted to conduct its own review of an ISIS strategy. "We would just take the step".
Still, Nauert sought to acknowledge Turkey's misgivings about terrorism by the PKK and other groups.
Sen. John McCain said Thursday that Turkey's ambassador to the United States should be removed after a clash between Turkish security and protesters in Washington D.C. this week. Turkey, the United States and European Union consider the PKK a terrorist organization.
Washington's Metropolitan Police Department said it had arrested two USA residents, Ayten Necmi, 49, and Jalal Kheirabadi, 42, over the incident, according to the BBC.
"Violence is never an appropriate response to free speech, and we support the rights of people everywhere to free expression and peaceful protest", a US State Department statement on Wednesday said.
The statement said that the USA communicated its concerns to the Turkish government "in the strongest possible terms".
On Wednesday Turkish Ambassador to the U.S. Serdar Kılıç was summoned to the State Department, a senior State Department official told CNN.
Republican Senator John McCain told MSNBC's "Morning Joe" on Thursday that the USA should throw "the ambassador the hell out of the United States". "The United States recognizes this inviolability, which provides reciprocal protection for the United States overseas". "Turkey is not a country that will consent to such treatment", Erdogan said in a speech to business leaders in Istanbul.