"Why would Kim Jong-Un insult me by calling me "old, ' when I would NEVER call him 'short and fat"?" he tweeted. Trump, 71, sarcastically responded that he hopes one day to be friends with Kim, who oversees a developing nuclear weapons and ballistic missile arsenal.
War of words is continued between Washington and Pyongyang for long.
The article also said Trump was a coward for cancelling a visit to the North Korea border, a decision that was made because of poor weather conditions.
North Korea is not known to have tested any of its missiles or nuclear devices since September 15, a relative lull after a brisk series of tests earlier this year.
In his speech, Trump congratulated South Korea's progress since the end of the 1950-53 Korean War, highlighting the differences in living standards between the two Koreas and personally addressed dictator Kim Jong Un, calling North Korea "hell" and urging him to give up his weapons.
Asked at news conference with Vietnam's president later Sunday if he could become friends with the North Korean leader, Trump replied it "might be a unusual thing to happen but it's certainly a possibility". "Every step you take down this dark path increases the peril you face".
Four dead, ten injured in California school shooting
There are just four teachers on staff; each teaches at least two grades, according to the school newsletter . The suspected shooter , who was not immediately identified, was killed by officers.
She said she was protesting against Mr Trump's attitude to women and his failure to meet with human rights activists in Vietnam. The post came in response to the country's foreign ministry calling Trump "an old lunatic" over his speech in South Korea.
Trump's descriptions of North Korea's current leader have varied, and he has even been positive at times - describing him as a "pretty smart cookie" in April. During the Asia Pacific Economic Cooperation summit in Danang, Vietnam, on Saturday, Trump and Russian President Vladimir Putin met privately.
An editorial in the ruling party newspaper Rodong Sinmun, focused its anger on Trump's visit to South Korea last week, during which he denounced Pyongyang's "cruel dictatorship", in a speech to legislators in the South.
On Twitter, Trump also slammed President Barack Obama and former secretary of state Hillary Clinton for failing to improve relations with Russian Federation.
As the president took to social media, three United States aircraft carriers were taking part in a military exercise in the Western Pacific, in a show of strength aimed at North Korea.