The key visit was marred by bloody clashes that took place outside the Turkish ambassador's residence after Erdogan met American counterpart Donald Trump.
US officials said Erdogan's bodyguards attacked a small group of pro-Kurdish protesters gathered outside the building and the incident ignited a storm of controversy.
Turkey blamed the violence on supporters of the outlawed Kurdistan Workers Party (PKK) who picked on pro-Erdogan US- based Turks.
The ministry summoned John Bass, American ambassador in Ankara, to deliver a "written and verbal protest" over "the aggressive and unprofessional actions taken."
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"It has been formally requested that the US authorities conduct a full investigation of this diplomatic incident and provide the necessary explanation," it added.
The ministry said it told Bass there were "lapses of security" during Erdogan's stay.
It hit out at "the inability of US authorities to take sufficient precautions at every stage of the official programme."
However, the ministry said it emphasised to Bass that this would not overshadow an otherwise "successful and important visit".
"This is Erdogan's security detail. Somebody told them to go out there and beat up on these peaceful demonstrators," McCain said Thursday.