In Tehran, a headline on a local newspaper quoted Rouhani as saying: "Mr. Trump, do not play with the lion's tail".
The White House on Monday said US President Donald Trump's threatening tweet shows he's not going to tolerate critical rhetoric from Iran, but claimed the US leader isn't escalating tensions between the two countries.
Iran's official government news agency IRNA said Trump was only "copying" Iranian Foreign Minister Mohammad Javad Zarif, who had reportedly warned Western countries during the 2015 nuclear negotiations to "never threaten an Iranian".
In all caps, Trump told Rouhani to never threaten the USA again or face historic consequences.
"Death to America" is a common refrain at rallies and on state-sponsored murals in Iran since the Islamic Republic was founded in 1979, and there was once a special holiday declaration of a "Death to America Day".
The deal with six world powers provided Iran with some relief from sanctions in return for curbs on its nuclear program.
"Never, ever threaten the United States again", the president admonished in an all-caps tweet. However, the language also echoes his volleys against North Korea's Kim Jong-Un, which has more recently evolved into diplomatic talks.
Trump has long-promised to take a hard line against Iran.
Israeli Prime Minister Benjamin Netanyahu praised the US presidents "strong stance" after years in which the Iranian "regime was pampered by world powers".
"To the regime, prosperity, security, and freedom for the Iranian people are acceptable casualties in the march to fulfill the Revolution", Pompeo said Sunday night at the Ronald Reagan National Library in Simi Valley, California.
The tweet came shortly after Secretary of State Mike PompeoMichael (Mike) Richard PompeoTrump warns Iran's Rouhani: Threaten us "and you will suffer" Pompeo: Iran's leaders resemble the mafia USA commander: Challenge with North Korea is making progress despite lack of trust MORE compared Iran's leaders to the mafia and touted U.S. support for the people of Iran, especially protesters who have taken to the streets recently in opposition to the current government.
He said despite poor treatment by their leaders, "the proud Iranian people are not staying silent about their government's many abuses", Pompeo said.
Iran's president also warned Trump against efforts to stop other countries from buying the Gulf nation's oil exports. "The United States supports you". "And he's gotten us all talking about Iran this morning, instead of talking about Putin, or the investigation into Russian interference". "By lodging personal attacks against Iran's supreme leader, making demands that no one believes Iran can fulfill, and hinting at regime change, the administration is making clear that military action is very much on the table".