The US President lashed out on social media in an apparent response to a televised speech delivered by Mr Rouhani hours earlier, in which he told the US “not to play with the lion’s tail”.
Mr Trump, writing in capital letters, posted:”To Iranian President Rouhani: never, ever, threatens the United States again or you will suffer consequences the likes of which few throughout history have ever suffered before.
“We are no longer a country that will stand for your demented words of violence & death. Be cautious.”
Speaking in Tehran on Sunday, Mr Rouhani had told Mr Trump not to escalate tensions with his country.
In language reminiscent of that used by Iraqi dictator Saddam Hussein in the lead-up to the Gulf War, he added: “Do not play with the lion’s tail or else you will regret it.
“Peace with Iran would be the mother of all peace and war with Iran would be the mother of all wars.
“You are not in a position to incite the Iranian nation against Iran’s security and interests.”
The United States has reimposed sanctions against the Islamic republic after Mr Trump pulled out of the JPCA nuclear deal aimed at preventing Iran from developing nuclear weapons, with Mr Trump saying:
“This was a horrible one-sided deal that should have never, ever been made.”
These sanctions penalise any country trading with Iran in various areas, and are aimed at preventing Mr Rouhani’s regime from exporting crude oil to the international market.
Earlier this month he hinted that Iran may seek to block regional oil exports if its own sales are hinted, and could potentially do so by preventing ships travelling through the Straits of Hormuz – although no mention has been made of this specifically up to now.
For his part, Mr Rouhani’s speech, especially the point about “inciting the Iranian nation” may have been in part a response to one by Mike Pompeo, the US secretary of state earlier on Sunday.
In it, he referred to Iran’s ruling elite as a “mafia”, claiming they had amassed huge sums of wealth while allowing their people to suffer.
Branding the county’s religious leaders “hypocritical holy men”, he referred to recent public pro-demoracy protests, adding: “Sometimes it seems the world has become desensitised to the regime’s authoritarianism at home and its campaigns of violence abroad, but the proud Iranian people are not staying silent about their government’s many abuses.”