The demand could foment a crisis in U.S.-Turkey relations, since the U.S. is unlikely to want to honor the request without hard evidence, which could affect Turkey's role as a key ally in the fight against ISIS and the handling of refugees fleeing the Syrian civil war.
Erdogan issued the call while addressing a crowd of supporters in Istanbul Saturday.
"I call on the United States and President Barack Obama: Dear Mr. President, I told you this before, either arrest Fethullah Gulen or return him to Turkey. You didn't listen," Erdogan told his backers.
Gulen, a former ally and cleric, now resides in exile in Pennsylvania. Erdogan blamed him almost immediately on Friday.
Secretary of State John Kerry told CNN's Jake Tapper Sunday that Turkey has made no formal request for extradition -- and that he'd asked the country's foreign minister to make the official request, saying that "the United States is not harboring anybody."
"We have always said, 'Give us the evidence, show us the evidence, we need a solid, legal foundation that meets the standards of extradition in order for our courts to approve such request,' " he told Tapper on "State of the Union."
"So we're waiting for that," Kerry said. "They tell us they are putting it together and will send it to us. But we think it's irresponsible to have accusations of American involvement when we're waiting on their request."
State Department spokesman John Kirby said late Saturday that Kerry had spoken with the Turkish foreign minister and during that call Kerry had offered U.S. assistance with an investigation into those involved with the coup. But he added that "public insinuations or claims about any role by the United States in the failed coup attempt are utterly false and harmful to our bilateral relations."
When asked about the possibility of Gulen's extradition in June, Kirby told reporters that, "We don't talk, as you know, about the specifics of extradition cases one way or the other."
The cleric has lived in the U.S. since 1999 and has never been charged with a crime during that time.
On Saturday, however, Erdogan further implored the U.S. in Istanbul: "I call on you again, after there was a coup attempt, extradite this man in Pennsylvania to Turkey! If we are strategic partners or model partners, do what is necessary."
Gulen, 75, denied the allegations in a statement Friday night, saying, "As someone who suffered under multiple military coups during the past five decades, it is especially insulting to be accused of having any link to such an attempt. I categorically deny such accusations."
In an interview on Saturday with Reuters, Gulen said he didn't know who was behind the coup: "I have been away from Turkey for 16 years. I have not been following these developments, therefore I cannot say anything about that."
Gulen also cast doubt on the prospect of his being extradited. "They have voiced such requests without making formal requests in the past," Gulen said prior to Erdogan's speech.
The situation risks developing into a major crisis in U.S.-Turkey relations if the U.S. declines to extradite or arrest Gulen, as it did with another nearby power.
A similar episode took place in 1979 when the U.S. refused to extradite the Shah of Iran, who was undergoing medical treatment in the U.S. at the time, after the revolution against his rule.
That refusal sparked outrage in Iran, which culminated in the storming of the U.S. Embassy in Tehran and the taking of American hostages.
The fallout from that affair has continued to impact U.S.-Iran relations to this day.
But the U.S. has also moved on exiles that were conspiring against the governments in their homelands.
Gen. Vang Pao, a Hmong military leader and major American ally during the Vietnam War, was arrested in 2007 for plotting to overthrow the Communist government in Laos. He was living in exile in California at the time of his arrest.
Erdogan seems intent on going after Gulen's supporters in the wake of the coup.
The Atlantic Council's Aaron Stein, who specializes in U.S.-Turkey relations, told CNN Friday that Erdogan had made cracking down on Gulenists a major priority for Turkey's internal security apparatus.
In an early warning sign, the Ankara chief public prosecutor's office took nearly 200 top Turkish court officials, including members of the supreme court, into custody, Anatolian News Agency reported Saturday.
Though Erdogan has frequently railed against and curtailed the judiciary for its supposed allegiance to Gulen, there has yet to be any evidence indicating that its members were behind the coup.
Erdogan has also accused Gulen of being behind a corruption scandal that engulfed many members of his government.
Gulen has a loyal following -- known as Gulenists -- in Turkey, who all subscribe to the Hizmet movement.
Hizmet is a global initiative inspired by Gulen, who espouses what The New York Times has described as "a moderate, pro-Western brand of Sunni Islam that appeals to many well-educated and professional Turks."
Gulen and Erdogan started out as political allies, both being opposed to Turkey's secular elite, but they had a falling out and Erdoan's government has since designated the Gulen movement a terrorist organization.
Kirby, however, said on Saturday that the U.S. does not consider Gulen's movement to be a foreign terrorist organization.
CNN's Amy La Porte, Ivan Watson and Gul Tuysuz contributed to this report.