Eshaq Jahangiri, Iran's first vice president, dismissed the remarks from Mattis as "useless claims," saying that Iran had been instrumental in the fight against ISIS, according to the state-run Islamic Republic News Agency, or IRNA.
"The Iranian government and nation do not care in the least for the worn-out and threadbare remarks made by the American officials," Jahangiri said.
Mattis said Iran's "misconduct and misbehavior" would have to be addressed.
"We have a responsibility with the rest of the nations to be absolutely clear with Iran in this regard. It does no good to ignore it. It does no good to dismiss it," Mattis said in Tokyo, where he met with his Japanese counterpart to discuss security issues.
But he said the recent tensions with Iran did not warrant an increase in the number of US forces in the Middle East.
"We always have the capability to do so, but at this time I don't think it's necessary," he said.
As Mattis made his comments Saturday, Iran's air force was conducting military drills, including missile systems, radar and electronic warfare command and control exercises, in the northern Semnan province, the semiofficial Fars news agency reported.
These drills are aimed at "showing strength, intelligence and readiness for all-round defense against any threat" and are being done "in contempt of sanctions and threats," the report said.
The commander of Iran's Islamic Revolutionary Guard Corps Aerospace Force spoke Saturday on drill sidelines.
"I'd say with confidence that foreign threats against the Islamic Establishment (of Iran) are futile," said Brig. Gen. Amir Ali Hajizadeh, quoted by the Tasnim News Agency.
He said US outcry over Iranian missile tests is a pretext for expressing hostility toward Iran.
"Should the enemy make a mistake, our roaring missiles will rain down on them," Hajizadeh said.
On Friday, the US Treasury Department said it was applying sanctions on 25 individuals and companies connected to Iran's ballistic missile program and those providing support to Iran's Islamic Revolutionary Guard Corps' Quds Force. That included three separate networks linked to supporting the missile program, which the United States opposes.
Trump tweeted Friday that Iran was "playing with fire" and he would not be as "kind" to Tehran as former President Barack Obama.
Iran's Ministry of Foreign Affairs said Friday it too would ensure "legal restrictions" were imposed on the "American individuals and companies which have a role in aiding extremist and terrorist groups or contribute to the suppression and murder of the defenseless people in the region," IRNA reported.
Trump has been a longtime critic of the accord, which was brokered after two years of talks with the five members of the UN Security Council and Germany in 2015.
Nasser Hadian, a professor of international relations at Tehran University, told CNN this week it was unlikely the Trump administration would tear up the agreement.
Abandoning the accord "would serve hard-line interests in Iran," he said.
CNN's Jennifer Deaton contributed to this report.
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Iran world's 'biggest state sponsor of terrorism,' Mattis says - CNN