In televised remarks, President Guillermo Lasso said Leonidas Iza, leader of the Confederation of Indigenous Nationalities, intends to "overthrow the government".
Lasso said he was willing to engage in talks on ending protests that have lasted nearly two weeks.
"It is proven that the true intention of the violent (people) is to generate a coup and that is why we call on the international community to warn of this attempt to destabilise democracy in Ecuador," Lasso said.
"Mr Iza can no longer control the situation. The violence perpetrated by infiltrated criminals has got out of hand."
The demonstrations are part of a national strike begun by the Confederation of Indigenous Nationalities on June 14 to demand fuel price cuts, cost controls on agricultural products and a larger budget for education.
Protests have been especially violent in six provinces in the north-central part of the South American country.
The confederation on Thursday said a demonstrator died of pellet wounds in the chest and abdomen while protesting near the National Assembly in Quito, where about 100 other people suffered a variety of injuries.
Police tweeted that officers were also injured by pellets.
Politicians from the Union por la Esperanza caucus, which is linked to former President Rafael Correa, on Friday began a legislative process seeking Lasso's removal.
It would take the votes of at least 92 lawmakers to remove Lasso, while Union has only 47 seats.
In Quito, protesters blocking roads have brought the city to a near standstill and people are experiencing food and fuel shortages.
Production Minister Julio Jose Prado said nearly 600 private vehicles and food-carrying trucks are trapped in Nanegalito, a community about 65km northwest of Quito.
Groups of protesters have roamed the city attacking vehicles and civilians and forcing the closure of businesses, some of which were looted. They have also punctured the tyres of buses, forcing passengers to walk.
Lasso urged Indigenous people and peasants "who have been brought to Quito with deception" to return to their communities for their own safety.
He added human rights groups should scrutinise the situation, which he said has violated the rights of security forces, citizens and journalists.
Human Rights International said four of its staff members were physically attacked and robbed on Thursday while "carrying out research and verification work on the protests" in Quito.
"We reject these actions, and we call for dialogue," the group tweeted.
The situation prompted several embassies, including those of Germany, Britain, Canada and the US, to issue a public statement expressing concerns about "the fundamental rights of all citizens".
