"To put my message on the same plane as what the president has attempted to do here, I think is not understanding what the message of the Gospel is," Leo told AP on Monday aboard the papal plane on the way to Algeria.
"And I'm sorry to hear that but I will continue on what I believe is the mission of the church in the world today."
History's first US-born pope stressed that he was not making a direct attack against Trump or anyone else with his general appeal for peace and criticisms of the "delusion of omnipotence" that is fuelling the Iran wars and other conflicts around the world.
"I will not enter into debate. The things that I say are certainly not meant as attacks on anyone. The message of the Gospel is very clear: 'Blessed are the peacemakers,'" Leo said.
"I will not shy away from announcing the message of the Gospel and inviting all people to look for ways of building bridges of peace and reconciliation, and looking for ways to avoid war any time that's possible."
Speaking to other reporters, he added: "I have no fear of the Trump administration."
Trump delivered an extraordinary broadside against Leo on Sunday night, saying he did not think the US-born global leader of the Catholic Church is "doing a very good job" and that "he's a very liberal person," while also suggesting the pontiff should "stop catering to the Radical Left".
Flying back to Washington from Florida, Trump used a lengthy social media post to sharply criticise Leo, then kept it up after deplaning, in comments on the tarmac to reporters.
"I'm not a fan of Pope Leo," he said.
Trump's comments came after Leo suggested at the weekend that a "delusion of omnipotence" is fuelling the US-Israel war in Iran.
While it's not unusual for popes and presidents to be at cross purposes, it's exceedingly rare for a pope to directly criticise a US leader - and Trump's stinging response is equally uncommon, if not more so.
"Pope Leo is WEAK on Crime, and terrible for Foreign Policy," the president wrote in his post, adding, "I don't want a Pope who thinks it's OK for Iran to have a Nuclear Weapon."
Later, Trump posted a picture suggesting he had saint-like powers akin to those of Jesus Christ.
Wearing a biblical-style robe, Trump is seen laying hands on a bedridden man as light emanates from his fingers, while a soldier, a nurse, a praying woman and a bearded man in a baseball cap all look on admiringly.
All of that came after Leo presided over an evening prayer service in St Peter's Basilica on Saturday, the same day the United States and Iran began negotiations in Pakistan during a fragile ceasefire.
The Pope did not mention the United States or Trump by name, but his tone and message appeared directed at Trump and US officials, who have boasted of US military superiority and justified the war in religious terms.
Leo - who is on an 11-day trip to Africa - has previously said God "does not listen to the prayers of those who wage war, but rejects them".
Before the ceasefire, when Trump warned of mass strikes against Iranian power plants and other infrastructure and that "an entire civilisation will die tonight", Leo described such sentiments as "truly unacceptable".
