But Turkey's allies have warned him against excessive retribution as the authorities round up the perpetrators.
Gulen's supporters say their group which they call Hizmet (Service) is entirely peaceful.
State media reported on Monday that more than 100 generals and admirals had been detained in raids across the country.
Erdogan has urged citizens to remain on the streets even after the defeat of the coup, in what the authorities describe as a "vigil" for democracy.
Erdogan supporters waving Turkish flags also thronged the central Taksim square in Istanbul - scene of mass anti-government protests three years ago - and a smaller crowd gathered outside the gates of the his vast presidential palace complex in the capital.
According to Anadolu, 1,800 additional elite special police forces have been drafted in from surrounding provinces to ensure security in Istanbul.
The detention came after Ankara launched an intensive crackdown on the judiciary and the military, with over 6,000 people reportedly arrested over the coup attempt.
U.S. officials were working with Turkish officials to resume air operations quickly, the Pentagon said.
"We are the ones saying today rule of law has to be protected in the country", she said in Brussels. "There can not be purges, the rule of law must work".
President Erdogan has been warned not to use the failed coup in Turkey as a "blank cheque" to bypass democratic principles.
"Just as I did not take any role in this attempted coup against our people and our democracy, I had done everything in my power to help our nation overcome the damage (from this coup)", the statement said according to Hurriyet. "We can not ignore this demand".
There has also been concern about the nature of the arrests which have appeared aimed at humiliating the suspects in the media.
Turkish television has shown images of captured suspects forced to lie face down on the tarmac after their arrest while AFP photographers have seen suspects roughly led away, pursued by angry mobs.
Anadolu published pictures of the arrest of former air force commander Akin Ozturk bent forwards, facing a wall with hands tied behind his back.
The two officials also discussed Turkey's importance as both a North Atlantic Treaty Organisation ally and a key member of the coalition to defeat the Islamic State (IS), while stressing the need to "stay focused on ensuring operational support to those efforts continue unabated".
Speaking to state-run news channel TRT Haber, Bekir Bozdag said keeping Mr Gulen "wouldn't befit the solidarity, co-operation, alliance and friendship between Turkey and the United States".
On Sunday, President Erdogan vowed to "clean all state institutions of the virus" of Fethullah Gulen supporters.
Gulen has categorically denied any involvement in the plot and suggested it could have been staged by Erdogan himself.
That much of the country, including those who have bitterly opposed his government, stood against a military coup as a violation of democracy has raised hopes that Erdogan will seize the moment to reach across Turkey's many political divides and unite the country.