Trump's action on Friday indicates he is willing to use his presidential powers as an economic and diplomatic weapon, even against allies, the Wall Street Journal reported.
Turkey has vowed to retaliate, but its currency has steadily weakened, removing some of the bite of the tariffs by making Turkish goods cheaper for US consumers.
Erdogan has cast the sell-off in the currency that has followed the trade row as an economic war and an attempt to undermine Turkey's economy.
The United States has denied Turkey's request for deporting Fethullah Gulen, a US -based Turkish cleric who Ankara says is the main perpetrator of the 2016 coup.
"Nevertheless, we implore President Trump to return to the negotiating table - this can and should be resolved through dialogue and cooperation".
On Friday, Trump issued a tweet on his official account saying he has just authorized a doubling of Tariffs on Steel and Aluminum with respect to Turkey as their currency, "the Turkish Lira, slides rapidly downward against our very strong Dollar!"
"The United States should know that the only result that such sanctions and pressure will bring. will be harming our relationship as allies", the foreign ministry said in a statement.
"If they [U.S. manufacturers] are importing from Turkey and they rely on them, then tariffs are going to make it more expensive for them by raising their own costs", said William Jackson, chief emerging markets economist at London-based Capital Economics.
Noting that Turkey is moving away from North Atlantic Treaty Organisation allies and toward Russian Federation and Iran, Mahaffee said, "The clash of personalities between Trump and Erdogan might be more than enough to split the historic alliance". "Relations with countries who behave like this have reached a point beyond salvaging", said Erdogan, who warned of "economic war".
Mr Erdogan said during an address to supporters: "Change the euros, the dollars and the gold that you are keeping beneath your pillows into lira at our banks".
The US was one of the top destination for Turkish steel exports in 2017, worth $1.1 billion.
Erdogan, a self-described "enemy of interest rates", wants cheap credit from banks to fuel growth, but investors fear the economy is overheating and could be set for a hard landing.
"This is a national, domestic battle".
In overseas trading, stock markets across the Asia-Pacific region moved mostly lower during trading on Friday.
"Investors said "Here we go again".
The currency's sharp decline has reflected concerns about the fundamentals of Erdogan's economic model, which has depended on a voracious construction industry that his opponents say has enriched his inner circle while heaping debt on the country.
New Finance Minister Berat Albayrak - Erdogan's son-in-law - has promised stronger budget discipline and a priority on structural reforms. As the currency drops, Turkish companies and households with debt in foreign currencies see their debts expand. It said the United States remained an important trade partner.