Erdogan told CNN in an exclusive interview Monday that the extradition request was coming soon.
Prime Minister Binali Yildirim announced the request Tuesday in Parliament and on Twitter referred to Gulen as a "terrorist leader."
The Muslim cleric, living in self-imposed exile in Pennsylvania, has denied any involvement in the coup attempt.
Gulen, in a statement released Tuesday, said Erdogan "once again demonstrated he will go to any length necessary to solidify his power and persecute his critics. It is ridiculous, irresponsible and false to suggest I had anything to do with the horrific failed coup. I urge the U.S. government to reject any effort to abuse the extradition process to carry out political vendettas."
Gulen, who lives in the United States, previously said any attempt to overthrow the country "is a betrayal to our unity and is treason."
Can Gulen be extradited?
Under the U.S.-Turkey extradition agreement, Washington can only extradite a person if he or she has committed an "extraditable act." Treason -- such as that implied by Erdogan's demand for Gulen's extradition -- is not listed as such an act in the countries' treaty.
When asked what evidence the government had that Gulen was behind the coup, Deputy Prime Minister Numan Kurtulmus said Tuesday that the attempt itself was the biggest piece of evidence, and that Turkey would provide thousands of pieces of evidence to the United States of Gulen's involvement.
He compared the coup attempt to the September 11, 2001, terror attacks, saying it was clear Gulen was behind it, just as the United States knew al Qaeda leader Osama bin Laden was behind 9/11.
Security forces gutted
The request for Gulen's extradition is the latest move by the Turkish government to rein in dissent in the country.
More than 9,300 people are in detention in the fallout from the failed coup, Kurtulmus said.
The government has gutted some of the security forces, dismissing almost 9,000 people from the Interior Ministry, mostly police officers, and hundreds of others from various ministries.
Among those detained are at least 118 generals and admirals, accounting for a third of the general-rank command of the Turkish military, according to Turkish state broadcaster TRT.
Some 15,200 public education employees were suspended and are being investigated for possible links to Gulen, the Ministry of Education said.
Authorities have revoked the licenses of 21,000 teachers working in private education institutions who are being investigated under the same auspices, state news agency Anadolu reported.
More than 80 judges are also among those arrested, as are lawyers, senior aides and police.
A photo emerged over the weekend of dozens of detainees, who appeared to be all men, seen stripped to the waist in a horse stable, their hands bound.
Asked about that, Kurtulmus said it was "normal procedure for police under these circumstances," adding that their "crime is very heavy."
Two pilots who downed a Russian jet last year are also among "the detained soldiers who attempted the coup," said Turkish Justice Minister Bekir Bozdag.
Erdogan continued to laud Turkish citizens for thwarting the plotters. "Our response to the coup attempt has proven that our democracy is strong," he said.
Friends turned foes
Erdogan and Gulen are former allies whose relationship fell into a bitter feud in 2013.
Erdogan supporters outside Gulen's Pennsylvania home have been calling him inflammatory names following the weekend violence. Gulen's supporters accused Erdogan of scapegoating the cleric to grab more power.
Gulen is a reclusive cleric who leads a popular movement called Hizmet, which includes hundreds of secular co-ed schools, free tutoring centers, hospitals and relief agencies credited with addressing Turkey's social problems.
Gulen supporters -- known as Gulenists -- describe the 75-year-old as a moderate Muslim cleric who champions interfaith dialogue.
WikiLeaks releases emails
Whistleblower site WikiLeaks
seems to think Turkey's purge has spread to cyberspace. It said it has come under a sustained cyberattack after announcing on social media its plan to leak hundreds of thousands of documents on "Turkish power."
The WikiLeaks website on Tuesday evening leaked 300,000 emails and thousands of documents in the wake of the failed coup.
The website appeared operational Tuesday, and WikiLeaks said it planned to go ahead with publishing the #ErdoganEmails, adding that all 300,000 were internal to Erdogan's Justice and Development Party.
The most recent emails were sent July 6 and the oldest dates back to 2010, the group said.
It was unclear when the other 500,000 documents would be released. The organization said the emails date up until July 7.
Death penalty talks
International pressure is mounting on Erdogan after he responded to the failed coup with an iron fist.
In the CNN interview Monday, he said refused to rule out the death penalty for the thousands arrested despite warnings from the European Union that reintroducing capital punishment would dash Turkey's chances of joining the the EU.
The EU official overseeing Turkey's bid to join, Johannes Hahn, expressed concern over Turkey's post-coup purge, raising suspicions that a list of people to arrest had been prepared well in advance of the political upheaval.
"(That) the lists are available already after the event indicates that this was prepared, that at a certain moment (they) should be used," he said.
U.S. Secretary of State John Kerry also warned that Turkey must respect the law and its democratic institutions if it wanted to remain part of NATO.