Brunson, an evangelical Christian pastor originally from North Carolina but a longtime resident of Turkey, was accused by the Turkish government of terrorist involvement and playing a role in the Turkish military's 2016 attempted coup to overthrow the government.
The lira crisis has deepened concerns about the broader economy - particularly Turkey's dependence on energy imports and whether foreign-currency debt levels pose a risk to the banking sector.
In U.S. markets, the Dow Jones Industrial Average .DJI rose 396.32 points, or 1.58 percent, to 25,558.73, the S&P 500 .SPX gained 22.32 points, or 0.79 percent, to 2,840.69 and the Nasdaq Composite .IXIC added 32.41 points, or 0.42 percent, to 7,806.52.
Turkey and the USA are now experiencing rocky relations after Washington imposed sanctions on two Turkish Cabinet ministers for not releasing Brunson.
Borrowing costs may rise further after both Moody's and Standard & Poor's ratings agencies cut Turkey's sovereign credit ratings deeper into "junk" territory late on Friday.More news: Aretha Franklin: Inside Her Walmart Workouts
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The country's officials would always support Chinese and Turkish companies signing cooperation projects based on market rules, he added.
After having clawed back most of its losses on Friday and Monday, when it shed 20 percent of its value against the dollar, the lira began falling again on the new sanctions threats, dropping by five percent at one point.
Turkey's banking watchdog has taken steps to stabilise the currency, limiting futures transactions for offshore investors and lowering limits on swap transactions.
The latest United States announcement came after Turkish Finance Minister Berat Albayrak tried to soothe the markets during an unprecedented conference call with hundreds of foreign investors, insisting Turkey would emerge "stronger" from the currency crisis and would not need an International Monetary Fund bailout. Economists gave Albayrak's presentation a qualified welcome and the lira subsequently held firm.
However, deep concerns remain about the potential for damage to the economy. Turkey is dependent on imports, priced in hard currency, for nearly all of its energy needs.