The letter says: "We were deeply saddened by the news of heavy casualties and injuries as a result of the terror attacks committed in Tehran".
The five men who carried out twin attacks in Tehran were Iranians who joined the Islamic State group in Iraq and Syria before returning last summer, the intelligence ministry said Thursday.
Iranian leaders sought to play down the attacks, with supreme leader Ayatollah Ali Khamenei saying: "These firecrackers that happened today will not have the slightest effect on the will of the people".
"They earlier left Iran and were involved in the crimes of the terrorist group in Raqqa and Mosul", the ministry said, referring to Islamic State's effective capital in Syria and a city it captured in Iraq.
"We underscore that states that sponsor terrorism risk falling victim to the evil they promote", Trump said in the statement.
The country's Supreme Leader said the attacks will add to the hatred that Iranians harbor toward the USA and Saudi Arabia.
The five gunmen had returned to Iran in August 2016 for a terrorist operation under the command of Abu Aysheh, a ranking commander of Daesh, with the objective of carrying out attacks in Iran's religious cities, the statement said. He concludes by threatening the royal family of Saudi Arabia and promises, "after Iran, it will be your turn".
ISIS, which adheres to a puritanical strain of Sunni Islam, considers Shias heretics and has carried out numerous attacks against Shia civilians, in Iraq in particular. Iran also has been fiercely opposed to the militant group.
Seoul won't change THAAD agreement with US: security adviser
Pyongyang is expected to demand the return of all four fishermen by accusing Seoul of enticing them to defect to the South. He also warned that North Korea could only face further global isolation and more economic difficulties.
A clearly furious Iran Foreign Minister Javad Zarif responded on Twitter Thursday, describing Trump's statement as "repugnant".
The Intelligence Ministry said its forces stormed several "safe houses" linked to the group in the country's northwest, according to state television.
State television also reported on Thursday that the death toll had risen to 16, citing Ahmad Shojaei, the head of the country's forensic centre.
They were the first claimed by Daesh in Iran. They never named the country directly, but the implication was clear.
Trump's comments brought criticism from Iranians on social media, who recalled their government's offers of support and the candlelight vigils held in Iran after the attacks of September 11, 2001 in the United States.
Two separate attacks shook Tehran on June 7. The siege lasted for hours, and one of the attackers blew himself up inside, according to Iran's state TV.
Iran's state broadcaster said a security guard was killed at the tomb and that one of the attackers was slain by security guards. A woman was also arrested. Police said a further five people were arrested around Khomeini's shrine on suspicion of involvement. He did not elaborate.
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