The State Department also says that it summoned Turkish Ambassador Serdar Kilic to discuss the altercation.
This was all happening while President Trump was hosting a visit with Erdogan in the White House.
Police in Washington broke up the melee, arrested two men and are pursuing charges against several others they are trying to identify with the help of the State Department and the Secret Service.
The U.S.is relying on regional allies including Turkey for intelligence-sharing and military assistance as it crafts a Syria policy, particularly as Iran and Russian Federation work to bolster Syrian President Bashar Assad's government.
Arizona Sen. John McCain, the Republican chairman of the armed services committee, called for the Turkish ambassador to the USA to be thrown out of the country on Thursday.
But underneath the talk of "a new era", a deep fissure remains over the Trump administration's decision to directly arm a Kurdish rebel group in Syria that Turkey labels a terrorist organization.
Video shows people pushing past police to confront a small group of protesters across the street in Sheridan Circle.
Turkey blamed the violence outside its ambassador's residence on demonstrators linked to the Kurdistan Workers Party (PKK), but Washington's police chief called it a "brutal attack" on peaceful protesters. "This isn't Turkey or a third world country", McCain said, speaking on MSNBC's "Morning Joe" program.
CPEC is not directed at India, says China
Discussions focused on China's New Silk Road trade project - an initiative to create new trading routes across parts of the developing world.
"The demonstrators began aggressively provoking Turkish-American citizens who had peacefully assembled to greet the President".
"We can not lift the rule of emergency unless everything reaches peace and welfare", Erdogan told a meeting of business leaders in Istanbul.
President Donald Trump walks Turkish President Recep Tayyip Erdogan to his vehicle following his visit to the White House in Washington, Tuesday, May 16, 2017.
In April, the USA had criticized Turkish airstrikes against Kurdish militants in Syria and Iraq.
"Customary global law affords heads of state and members of their entourage with inviolability from arrest and detention", the official said.
The United States said it summoned Turkey's ambassador to the State Department, where the No. 2-ranked USA diplomat raised concerns about the security detail for President Recep Tayyip Erdogan's visit to Washington, after they were captured on video violently breaking up a protest.
President Recep Tayyip Erdogan angrily told Washington at the time to choose between Ankara and "terrorists".