The conflict, which began with joint Israeli-US strikes on Iran at the end of February, has killed more than 2000 people, disrupted the lives of millions of others and shaken energy and financial markets across the globe.
In a post on social media late on Friday, Trump said the US was "totally destroying the terrorist regime of Iran".
"We have unparalleled firepower, unlimited ammunition, and plenty of time - Watch what happens to these deranged scumbags today," Trump said.
"They've been killing innocent people all over the world for 47 years, and now I, as the 47th President of the United States of America, am killing them. What a great honor it is to do so!"
New Iranian Supreme Leader Mojtaba Khamenei, in his first comments on Thursday, vowed to keep the key energy choke point of the Strait of Hormuz shut and called on neighbouring countries to close US bases on their territory or risk Iran targeting them.
"I assure everyone that we will not neglect avenging the blood of your martyrs," said the hardline cleric, who replaced his father after he was killed in Israel's initial strikes.
His comments were read out by a television presenter and it was not clear why he did not appear in person.
Iranian officials have said he had been lightly wounded and Trump has said he thinks he is alive but "damaged".
Israeli Prime Minister Benjamin Netanyahu also held his first news conference since the US and Israeli air strikes on Iran started on February 28, issuing a veiled threat to kill new Khamenei and defending the military assault.
"I will not detail the actions we are taking. We are creating the optimal conditions for toppling the regime but I won't deny that I can't tell you with all certainty that the people of Iran will topple the regime - a regime is toppled from the inside," Netanyahu said on Thursday.
"But we can definitely help and we are helping."
The Israel Defence Forces said on Friday its air force struck more than 200 targets in western and central Iran in the past day, including ballistic missile launchers, air defence systems and weapons production sites.
Iran launched its own barrage of missiles and drones overnight, undermining earlier US and Israeli claims to have knocked out much of Iran's stock of long-range weapons.
An Iranian missile struck a Bedouin Arab town in northern Israel near Nazareth, according to the military, heavily damaging several homes.
Israel's ambulance service said 58 people were injured.
In Iraq, US Central Command said it was carrying out rescue efforts after one of its refuelling aircraft went down in an incident that involved another aircraft but was not the result of hostile or friendly fire.
In central Dubai, debris from a successful interception also caused minor damage to the facade of a building, the emirate's media office said early on Friday.
Two tankers were set ablaze in the Iraqi port of Basra earlier this week after being hit by suspected Iranian explosive-laden boats and other ships have been struck in the Gulf near the Strait of Hormuz, which has been effectively closed and through which a fifth of global oil usually passes.
Drones and missiles have also regularly struck buildings and other infrastructure in Kuwait, Saudi Arabia, the United Arab Emirates, Bahrain and Oman.
The prospect that one of the most severe disruptions ever to global energy supplies could endure sent oil prices up to $US100 a barrel on Thursday.
In an effort to stabilise global energy markets, the US on Thursday issued a 30-day licence for countries to buy Russian oil and petroleum products stranded at sea.
Trump, who has already declared that the US and Israel won the war, and his administration have given conflicting reasons for starting the fight, and have not provided a public assessment of its expected cost or duration.
