Trump has repeatedly criticised the religious leader, who has emerged as an outspoken critic of the US-Israeli war on Iran in recent weeks.
Trump says he is responsible for Leo's appointment, and has accused him of being "weak" on foreign policy.
"The Pope has to understand - it's very simple - Iran cannot have a nuclear weapon. The world would be in great danger," he told reporters at the White House on Thursday.
Trump's criticism has triggered widespread backlash from US Christians across the political spectrum.
Leo, originally from Chicago, decried the "madness of war" in a peace appeal on Saturday.
The US and Israel attacked Iran on February 28. Iran responded with its own strikes on Israel and Gulf states with US bases. US-Israeli strikes on Iran and Israeli attacks in Lebanon have killed thousands and displaced millions.
The Pope told Reuters on Monday he would keep criticising the war, regardless of Trump's comments.
"The Pope can say what he wants, and I want him to say what he wants, but I can disagree. I think that Iran cannot have a nuclear weapon," Trump said.
The Archbishop of Canterbury, Sarah Mullally, threw her support behind Pope Leo on Thursday, urging 85 million Anglicans worldwide to speak out for peace after Trump's attack.
In her first public comments addressing Trump's criticism, the leader of the Church of England signalled her solidarity with Leo, who has emerged as an outspoken critic of the war in recent weeks.
"I stand with my brother in Christ, His Holiness Pope Leo XIV, in his courageous call for a kingdom of peace," Mullally, the spiritual head of the Anglican Communion, said in a statement.
"As innocent people are killed and displaced, families torn apart, and futures destroyed, the human cost of war is incalculable. It is the calling of every Christian – and of all people of faith and goodwill – to work and pray for peace."
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