In a live televised address, Mr Rouhani branded the sanctions against Ayatollah Ali Khamenei and other senior officials, including Iran’s foreign minister Mohammad Javad Zarif, as “outrageous and idiotic”.
"You sanction the foreign minister simultaneously with a request for talks," he said, adding: "The White House is afflicted by mental disability and does not know what to do."
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Iran had earlier warned that Monday’s sanctions meant the permanent closure of diplomatic channels between Tehran and Washington.
Foreign ministry spokesman Abbas Mousavi said the American government was “destroying the established international mechanisms for maintaining world peace and security”.
Tensions high as Trump approves new Iran sanctions: In pictures
Show all 17
Tensions high as Trump approves new Iran sanctions: In pictures
1/17
Iran claims that in this picture released by Iran state TV, their surface-to-air missile is seen as it shoots down a US surveillance drone
EPA
2/17
This photo shows US RQ-4A Global Hawk unmanned surveillance drone. A drone of this model was shot down by Iran on Thursday 21 June
AFP/Getty
3/17
An oil tanker is on fire after it was subject to a suspected attacked at the Gulf of Oman on June 13. The US has blamed Iran for the attack
Reuters
4/17
Iran claims that in this picture released by Iran state TV, debris from the downed US drone is seen after it was recovered from Iranian waters
AFP/Getty
5/17
Smoke billows from an oil tanker after it was subject to a suspected attacked at the Gulf of Oman on June 13
Reuters
6/17
US President Trump holds up a signed executive order to increase sanctions on Iran on 24 June
AP
7/17
Iranian President Rouhani stated in a televised address that the White House is "afflicted by mental retardation" following the increase in sanctions on 25 June
EPA
8/17
An Iranian navy boat tackles the fire on the Norwegian owned Front Altair oil tanker after it was hit in a suspected attack
AFP/Getty
9/17
Iran claims that in this picture released by Iran state TV, debris from the downed US drone is seen after it was recovered from Iranian waters
AFP/Getty
10/17
Smoke billows from an oil tanker after it was subject to a suspected attacked at the Gulf of Oman on June 13
Reuters
11/17
A screenshot from a video released by the US Department of Defense that the US claims to show Iranian removing an unexploded limpet mine form the hull of the Japan-owned ship that was attacked in the Gulf of Oman on June 13
Getty
12/17
An item reportedly found on the Japan-owned oil tanker that was attacked on June 13 in the Gulf of Oman
AFP/Getty
13/17
A handout photograph from the US Department of Defense shows a composite material that the US claim was left behind on the hull of the Japan-owned oil tanker following the removal of an unexploded limpet mine
Getty
14/17
A view from the cabin of an Iranian navy boat as it tackles the fire on the Norwegian owned Front Altair oil tanker after it was hit in a suspected attack
EPA
15/17 Release by US government
A picture released by U.S. Central Command shows damage to the hull of the oil tanker Kokuka Courageous. The picture suggests that the ship is 'likely' to have been hit by a mine as the US Secretary of State Mike Pompeo moves to blame Iran for the suspected attack
Reuters
16/17
Smoke billows from an oil tanker after it was subject to a suspected attacked at the Gulf of Oman on June 13
Reuters
17/17 Release by US government
A picture released by U.S. Central Command shows damage to the hull of the oil tanker Kokuka Courageous. The pictures suggests that the ship is 'likely' to have been hit by a mine as the US Secretary of State Mike Pompeo moves to blame Iran for the suspected attack
EPA
1/17
Iran claims that in this picture released by Iran state TV, their surface-to-air missile is seen as it shoots down a US surveillance drone
EPA
2/17
This photo shows US RQ-4A Global Hawk unmanned surveillance drone. A drone of this model was shot down by Iran on Thursday 21 June
AFP/Getty
3/17
An oil tanker is on fire after it was subject to a suspected attacked at the Gulf of Oman on June 13. The US has blamed Iran for the attack
Reuters
4/17
Iran claims that in this picture released by Iran state TV, debris from the downed US drone is seen after it was recovered from Iranian waters
AFP/Getty
5/17
Smoke billows from an oil tanker after it was subject to a suspected attacked at the Gulf of Oman on June 13
Reuters
6/17
US President Trump holds up a signed executive order to increase sanctions on Iran on 24 June
AP
7/17
Iranian President Rouhani stated in a televised address that the White House is "afflicted by mental retardation" following the increase in sanctions on 25 June
EPA
8/17
An Iranian navy boat tackles the fire on the Norwegian owned Front Altair oil tanker after it was hit in a suspected attack
AFP/Getty
9/17
Iran claims that in this picture released by Iran state TV, debris from the downed US drone is seen after it was recovered from Iranian waters
AFP/Getty
10/17
Smoke billows from an oil tanker after it was subject to a suspected attacked at the Gulf of Oman on June 13
Reuters
11/17
A screenshot from a video released by the US Department of Defense that the US claims to show Iranian removing an unexploded limpet mine form the hull of the Japan-owned ship that was attacked in the Gulf of Oman on June 13
Getty
12/17
An item reportedly found on the Japan-owned oil tanker that was attacked on June 13 in the Gulf of Oman
AFP/Getty
13/17
A handout photograph from the US Department of Defense shows a composite material that the US claim was left behind on the hull of the Japan-owned oil tanker following the removal of an unexploded limpet mine
Getty
14/17
A view from the cabin of an Iranian navy boat as it tackles the fire on the Norwegian owned Front Altair oil tanker after it was hit in a suspected attack
EPA
15/17 Release by US government
A picture released by U.S. Central Command shows damage to the hull of the oil tanker Kokuka Courageous. The picture suggests that the ship is 'likely' to have been hit by a mine as the US Secretary of State Mike Pompeo moves to blame Iran for the suspected attack
Reuters
16/17
Smoke billows from an oil tanker after it was subject to a suspected attacked at the Gulf of Oman on June 13
Reuters
17/17 Release by US government
A picture released by U.S. Central Command shows damage to the hull of the oil tanker Kokuka Courageous. The pictures suggests that the ship is 'likely' to have been hit by a mine as the US Secretary of State Mike Pompeo moves to blame Iran for the suspected attack
EPA
Mr Trump said the “hard-hitting” sanctions, which came following weeks of escalating tensions between the two countries, were “a strong and proportionate response to Iran’s increasingly provocative actions”.
The president’s national security adviser, John Bolton, an Iran hawk with a history of calling for military action against the Islamic republic, has insisted Washington is still willing to talk to Tehran.
“The president has held the door open to real negotiations to completely and verifiably eliminate Iran’s nuclear weapons programme, its ballistic missile delivery systems, its support for international terrorism and other malign behaviour worldwide,” he said.
“All that Iran needs to do is to walk through that open door.”
Tensions have been mounting in the Gulf since Mr Trump withdrew the US from a 2015 multilateral deal in which Iran agreed to limit its nuclear programme in return for the lifting of crippling economic sanctions.
Those tensions culminated in the shooting down of a US drone last week. Mr Trump said he ordered reprisal air strikes but called them off at the last minute when an adviser told him it would result in an estimated 150 deaths.
The US has also blamed Iran for attacks on two oil tankers this month near the Strait of Hormuz, though officials have so far provided only shaky evidence for the claim, which Iran denies.
Donald Trump: ‘Lets make Iran great again’
According to the IRIB news agency, foreign minister Mohammad Javad Zarif said on Tuesday Iran would never pursue a nuclear weapon.
He pointed to the past use of nuclear weapons by the US and to Mr Trump's comment about wanting to avoid the deaths of 150 people.
"You were really worried about 150 people? How many people have you killed with a nuclear weapon? How many generations have you wiped out with these weapons?" Mr Zarif said.
"It is us who, because of our religious views, will never pursue a nuclear weapon," he added.
Washington claimed Monday's sanctions measures effectively freeze billions of dollars worth of Iran assets, but experts said the true impact was likely to be limited.
Richard Johnson, the US state department’s former acting assistant coordinator of the Iran nuclear deal, told BBC Radio 4’s Today programme: “Frankly I doubt that there’s many international companies doing business with these organisations, so I don’t think the sanctions will have a really substantive effect on Iran’s economy.
“I do think it’s much more symbolic than it is substantive.”
On Tuesday the UK, Germany and France issued a joint statement calling for "de-escalation and dialogue" amid "grave concern" about increased tensions in the Gulf.
They also reiterated their commitment to the 2015 nuclear deal, and urged "all sides to do the same".
"It is in everyone’s interest to show restraint and avoid any actions that would undermine this vital pillar of the non-proliferation regime and of our collective security," the statement said.
The original version of this article quoted Iranian president Hassan Rouhani as describing the White House as being "afflicted by mental retardation," based on a translation from Reuters. This has now been updated to include a more specific translation.
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