Pompeo said America stands in solidarity with Iranians and reiterated that the US expects countries to "significantly" reduce their dependence on Iranian oil or face tough new sanctions after November 4.
US President Donald Trump has launched a furious all-caps Twitter rebuke of Iran declaring "you will suffer consequences the likes of which few have ever suffered before".
President Donald Trump withdrew the USA from the Iran nuclear deal in May and the first sanctions to be reimposed were expected to hit on August 4.
Rouhani cautioned Trump yesterday about pursuing hostile policies against Tehran.
Trump has been a vocal critic of Iran since before he took office, vowing to scrap the Obama-negotiated nuclear deal - which he did in May, despite allies including the United Kingdom highlighting its success so far.
Mr Rouhani suggested peace was still possible and said: "America must understand well that peace with Iran is the mother of all of peace, and war with Iran is the mother of all wars".
"Those who fear this insane president's psychological warfare will know that America will not be content with anything less than our annihilation", said Gheypour.
Trump took to Twitter to reply, as he tends to do. "The United States is with you", he said.
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They shared the recording with Trump's lawyers as well, The New York Times reported , citing three people briefed on the matter. He added that a lawyer secretly recording a client was "totally unheard of and perhaps illegal".
The other nations in the agreement - Britain, France, Germany, Russia, and China - unsuccessfully urged Washington to remain a part of the deal, saying it was the best way to prevent Iran from obtaining nuclear weapons.
He lit into what he called Iran's "hypocritical holy men", saying the ruling elites have enriched themselves through corruption, and called out officials by name who he said had plundered government coffers through embezzlement or by winning lucrative contracts.
One high-ranking Iranian officer claimed Trump's overnight outburst was nothing but "psychological warfare", according to state TV.
The secretary also announced that the Broadcasting Board of Governors, which oversees the Voice of America, is taking new steps to help Iranians get around internet censorship.
He said "overthrow, disintegration, boycott and weakening" were the main USA policies against Iran.
"The decision for regime change is one that should come from the people of Iran, and they will decide to do this, whether it comes sooner or later".
Iran's Supreme Leader Ayatollah Ali Khamenei on Saturday backed Rouhani's suggestion that Iran may block Gulf oil exports if its own exports are halted.
"Such policies will unite Iranians who will overcome plots against their country", Iranian Foreign Ministry spokesman Bahram Qasemi said. He assured them that the Trump administration shared their dreams for the people of Iran. The United States supports you.
Trita Parsi, founder and president of the lobby group National Iranian American Council, said Pompeo's tough talk could fuel grievances, destabilise Iran and set Tehran and Washington on a collision course. Iranian leaders reject U.S. claims about developing nuclear weapons and supporting terrorism.