He was remanded in custody over "terror charges" by an Istanbul court last week.
Gulen's movement is also accused of orchestrating a series of corruption raids against high-ranking government officials in 2013.
Topuz is also accused of being in contact with four police officers implicated in the coup investigations. Former minister Zafer Caglayan "has protected Turkey's interests as Turkish economy minister, and has acted within the laws of our country and global laws while doing that", government spokesman Bekir Bozdag had said at the time.
The US embassy said on Thursday that it was "deeply disturbed" over the arrest and rejected the allegations against the employee as "wholly without merit". He did not deny the reports.
Enraged over the arrest, the United States authorities halted visa services and later the Turkish government responded in the same way.
"Trying a Turkish citizen for a crime committed in Turkey is our right".
"This looks like a really serious situation", said Blue Bay Asset Management strategist Timothy Ash, quoted by Reuters on October 9.
"The interlocutor for the U.S. embassy is our Washington embassy. But we believe the risk of a downward spiral in [lira] can not be ruled out - this will depend on diplomatic developments". "We are approaching this with the utmost rationality".More news: Google Pixel 2 turns on 'DND' mode while driving
The U.S. move comes at a moment when relations between the two allies are going through severe turbulence. No reason was cited for the cancellation.
State media named locally-hired Metin Topuz as the staff member arrested last week.
According to Mr Barkey, the Turkish president is interested in retaining those Turks arrested for violating the Iran sanctions. Turkish Airlines, the country's national carrier, saw its shares fall by 8%.
The two countries' diplomatic ties have been fraught lately, hitting a low point in May with an incident involving Turkish President Recep Tayyip Erdogan's security detail assaulting protestors outside the Turkish ambassador's residence in Washington, DC.
Since the failed coup, Turkish officials have worked to rid the government, educational system and military of suspected Gülen supporters. A translator at another American consulate in Turkey was arrested in March on suspicion of links to an outlawed Kurdish insurgent group.
Other Americans held in Turkey include Pastor Andrew Brunson who was detained past year and is accused of terror-related charges for alleged ties to Gulen's movement.
Emerging-market currency volatility has climbed since reaching an nearly three-year low in August, and exceeded expected fluctuations in developed-nation peers last week for the first time since July, JPMorgan Chase & Co. indexes show.
The action weighed heavily on the lira and Turkish stock markets in already thin trading.