"Iran must never be allowed to possess a nuclear weapon," he told reporters in Jerusalem, speaking beside Israeli President Reuven Rivlin.
He flew in from Saudi Arabia, a key US ally, where he gave a speech to Arab and Muslim leaders at a summit.
Mr Trump is holding separate talks with both Israeli and Palestinian leaders.
He has called an Israeli-Palestinian peace agreement "the ultimate deal" but has been vague about what form it should take, saying he prefers to leave it to both sides to decide between them in direct talks.
The two-day visit to Israel forms part of Mr Trump's first foreign trip as US president.
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What else did Trump say about Iran?
It must also "cease its deadly funding, training and equipping of terrorists and militias", Mr Trump said at President Rivlin's residence.
In a deal with world powers in 2015, Iran accepted curbs on its nuclear programme in return for tangible economic benefits, and the White House confirmed last month that the deal was still holding.
Military trucks carrying Iranian S300 missiles on parade in Tehran, 18 AprilImage copyrightEPA
Image caption
Iran showed off its air defence missiles at a parade last month
However, US Secretary of State Rex Tillerson has underlined that his country's policy towards Iran is under full-scale review. Israel, meanwhile, considers Iran a mortal threat.
Speaking later to Israeli Prime Minister Benjamin Netanyahu in Jerusalem, Mr Trump pledged Iran would never get nuclear weapons.
Iranian President Hassan Rouhani, who was re-elected for a second term last week, championed the 2015 deal but on Monday he appeared to sweep aside international concern about Iran's missile programme.
"The Iranian nation has decided to be powerful," he said on state TV. "Our missiles are for peace and for defence... American officials should know that whenever we need to technically test a missile, we will do so and will not wait for their permission."
Mr Rouhani also played down Mr Trump's strong criticism of Iran at a summit in Saudi Arabia at the weekend, saying: "Who can say regional stability can be restored without Iran?"
Where else did Trump go in Jerusalem?
After leaving President Rivlin's residence, he visited the Church of the Holy Sepulchre where, according to Christian tradition, Jesus was buried and resurrected.
Then, wearing a Jewish skullcap as a mark of respect, he visited the Western Wall, one of the most sacred sites in Judaism.
Donald Trump touches the Western Wall in Jerusalem, 22 MayImage copyrightEPA
The wall is a remnant from the time of the Second Jewish Temple, destroyed by the Romans in 70 AD.
Mr Trump could be seen placing what appeared to be a written note between its stones, as is the custom among visitors to leave notes there, such as requests, to God.
Can Trump's trip bring peace any closer?
Israel and the Palestinians have not held direct talks in just over three years.
Mr Trump is due to meet Palestinian leader Mahmoud Abbas in Bethlehem on Tuesday.
Just before Mr Trump arrived, Israel announced economic and development concessions for Palestinians, including easing some restrictions on movement and approving industrial construction projects.
The US president has been widely seen as considerably more supportive of Israel than his predecessor, Barack Obama, taking a softer position on the contentious issue of Israeli settlements.