Iran says the US has “crossed a red line” by imposing sanctions on the head of its judiciary, Ayatollah Sadeq Amoli-Larijani.
The foreign ministry vowed to retaliate, but did not say what form any action might take.
US President Donald Trump, a critic of the 2015 accord, said on Friday he would extend sanctions relief over the landmark deal one last time.
At the same time the US imposed fresh sanctions on 14 individuals and entities over alleged rights abuses.
President Trump said he was giving Europe and the US “a last chance” to fix “terrible flaws” in the nuclear agreement signed by Iran and six world powers in 2015.
The White House wants EU signatories to agree permanent restrictions on Iran’s uranium enrichment. Under the current deal they are set to expire in 2025. Mr Trump also wants Iran’s ballistic missile programme to be addressed.
Although the US has suspended sanctions against Iran following the nuclear deal, it still imposes punitive measures over issues such as terrorism, human rights and ballistic missile development.
The US Treasury issued a statement on Friday saying Ayatollah Amoli-Larijani was responsible for the “torture or cruel, inhumane, and degrading treatment or punishment of prisoners in Iran, including amputations”.
He had called for a crackdown on “rioters” and “vandals” following a recent spate of anti-government protests in Iranian cities.
Iran has complained that US sanctions not connected to nuclear activity have effectively cancelled out any financial benefits it expected from the 2015 deal.