Three women protesters were thrown out of the venue by Turkish security. An Ecuadorean MP was also reportedly injured during Thursday's incident.
Ecuador's foreign minister says the guards' behaviour was "irresponsible".
Mr Erdogan was in Quito to boost diplomatic and trade ties with Ecuador.
As Mr Erdogan prepared to hold a news conference at the National Higher Studies Institute, protesters demonstrated against recent operations by Turkey against the Kurdistan Workers's Party (PKK) militant group.
Inside the hall, when the three women heckled Mr Erdogan, they were set upon by his personal guards and a scuffle ensued. One protester was momentarily placed in a headlock.
According to local media, a member of Mr Erdogan's security entourage attacked a protester as the Turkish president left the building.
And a member of the Ecuadorean National Assembly was reportedly injured while trying to protect the protesters.
Lawmaker Diego Vintimilla posted images on his Twitter account showing himself with cuts to his nose and hands and a bruise on his arm.
Ecuador said disproportionate force had been used by the security guards.  Deputy Foreign Affairs Minister Fernando Yepez has summoned the Turkish ambassador in Quito, urging him to explain the violence.
Ecuadorean Interior Minister Jose Serrano said police had requested that the bodyguards involved in the incident surrender their passports on Friday.
However, Mr Erdogan and his entourage have since left the country. No comments from his government over the incident have been reported.
The PKK is considered a terrorist group by Turkey and its Western allies and has fought a 30-year insurgency in south-eastern Turkey in which tens of thousands of people have been killed.
There have been repeated clashes between PKK separatists and the Turkish army in recent months, and the violence has recently escalated.
Turkey has previously dismissed claims by the PKK that the government is attacking it "to stop the Kurdish advance against Islamic State [in Syria]".
Ecuador has protested to Turkey over an incident in which demonstrators were violently ejected during a speech by visiting Turkish President Recep Tayyip Erdogan in the capital, Quito.