By Agence France-Presse
Trump's high-stakes visit to the region was overshadowed by what the president described as a "horrific shooting" in Texas, where a gunman killed at least 26 people and wounded 20 more during Sunday services.
"Our hearts are broken," he told business tycoons in Tokyo.
Trump began his marathon trip in belligerent form, warning on Sunday that "no dictator" should underestimate US resolve, a clear swipe at North Korea and its young leader Kim Jong-Un.
However he then appeared to open the door for talks with Pyongyang, saying in an interview broadcast on US TV that he would "certainly be open" to meeting Kim.
"I would sit down with anybody," he said. "I don't think it's strength or weakness, I think sitting down with people is not a bad thing," he said on the "Full Measure" show.
"So I would certainly be open to doing that but we'll see where it goes, I think we're far too early."
The president arrived in Asia with tensions over North Korea at fever pitch, as US bombers fly sorties over the Korean peninsula and concerns mount that Pyongyang might stage another nuclear or missile test.
'Really extraordinary' relationship
Hawkish Abe and Trump are closely allied on the North Korea issue, with the Japanese prime minister sticking firmly to the US line that "all options" are on the table to deal with the rogue state.
"The closeness of the relationship is unprecedented. And the degree to which US and Japanese strategies are aligned, both on the Korean Peninsula but also throughout the Indo-Pacific, is also unprecedented," a senior Trump administration official said Sunday.
That closeness was demonstrated on the first day, when the two golfing buddies hit the course within hours of Trump touching down on Japanese soil.
Both leaders later tweeted images of their nine-hole encounter, with Abe saying the "relaxed" nature of their game allowed them to have "candid" discussions on some "difficult" issues.
However, that did not stop Trump lashing out at the trade relationship with Japan on Monday, saying it had been "winning" for decades at the expense of the United States.
"We want fair and open trade but right now our trade with Japan is not fair and open," Trump told business leaders.
"The US has suffered massive trade deficits with Japan for many, many years. So we will have to negotiate and we will do this in a friendly way."
Focus on human rights
Trump will on Monday meet families of civilians kidnapped by North Korea to train their agents in Japanese culture and language -- an emotional issue that still angers Tokyo.
North Korea has admitted to kidnapping 13 Japanese civilians but Japan believes the regime took dozens more, including a 13-year-old girl abducted on her way home from school.
"You are going to see some focus on the often overlooked question of human rights conditions in NK," said the White House official.
Trump stressed US military power and resolve while wearing a bomber jacket in front of service personnel when he landed in Japan, but speaking to reporters on the plane, he reserved some warm words for the North Korean people.
"I think they're great people. They're industrious. They're warm, much warmer than the world really knows or understands. They're great people. And I hope it all works out for everybody," he said.
However, he described North Korea as "a big problem for our country and for our world and we want to get it solved".
Trump also announced that he would hold talks with Russian President Vladimir Putin during his trip, as the global community scrambles for a solution to the North Korean crisis.
"We want Putin's help on North Korea," he said.
After meeting the Emperor and Empress of Japan, Trump holds talks with the prime minister and will brief the media before a final banquet with his "friend" Abe.
But Trump's Asian tour is set to become more testing as he flies to South Korea, then on to China.
Trump has a cooler relationship with South Korean President Moon Jae-In, whose dovish approach to the crisis he has denounced as "appeasement".
The president will address the parliament in Seoul but will not visit the Demilitarized Zone dividing the Korean peninsula -- a visit derided in Washington as a bit of a "cliche."
From Seoul, Trump travels to Beijing to meet his counterpart Xi Jinping who has solidified his grip on power after being handed a second term.