Erdogan said he warned Trump that Turkey would combat YPG if the group posed any security threat.
On Wednesday, the State Department said "violence is never an appropriate response to free speech".
"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.
Turkey's foreign minister also demanded that a U.S. envoy be removed for allegedly backing the Kurds, but the State Department said Brett McGurk has the "full support" of the Trump administration. Nine people were hurt.
The attack allegedly perpetrated by bodyguards and other supporters of Turkish President Recep Tayyip Erdogan came under fierce criticism from Washington, D.C., police and local officials, who described it as a violent attack on peaceful demonstrators.
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The US respects Turkish concerns about its "by, with, and through" approach to the Syrian Democratic Forces led by the YPG, and will continue consulting with ankara as the focus on combating IS continues, Nauert said.
The clash happened as President Recep Tayyip Erdogan returned to the embassy after meeting with President Donald Trump earlier Tuesday.
Washington Mayor Muriel Bowser said it was "an affront to DC values and our rights as Americans".
"We should throw the Turkish ambassador out of the country, we should identify those people that performed these unlawful acts of beating people up and they should be charged", Senator John McCain said on national television on Thursday morning.
The security detail, he continued, violated American laws on United States ground, and that can not happen.
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Metropolitan Police Department Chief Peter Newsham condemned the action Wednesday as an "assault" on those protesting "peacefully", warning that the department will "pursue everything that's within our legal power to hold the folks that were responsible accountable for their actions".
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.
Cavusoglu said Turkey received USA assurances that arms sent to the YPG would be used only against IS, without explaining how this would be monitored. Kheirabadi said he is a Kurd who passed through Turkey as a refugee and is now a USA citizen.
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.
Erdogan responded that there is there is no place for any Kurdish "terrorist organizations" in any agreement about the region's future.
Rain on the increase this weekend — FIRST ALERT FORECAST
Sunday night cloud cover builds back in and lows will be in the upper 50s with east winds at 5-10 miles per hour . We have a very energized atmosphere right now which is prompting the development of scattered showers and storm.