North Korea has branded Japan's PM Shinzo Abe an "imbecile" and "political dwarf", and accused him of mislabelling its latest weapons test.
Mr Abe condemned the North for "repeated launches of ballistic missiles" after two projectiles were fired on Thursday.
But the North insisted it was testing a "super-large multiple-rocket launcher".
On Saturday, state media said Japan "may see what a real ballistic missile is in the not distant future".
North Korea is banned from firing ballistic missiles under UN Security Council resolutions.
The country is under various sets of sanctions over its missile and nuclear programmes. Lifting the sanctions has been a key aim of the North in talks with the US – Japan's ally – but these have stalled since a summit between its leader Kim Jong-un and President Donald Trump broke down in February.
What's the latest spat about?
The North fired what observers in South Korea called two "unidentified projectiles" from its South Hamgyong province into the Sea of Japan on Thursday.
Condemning the launch, Mr Abe said: "North Korea's repeated launches of ballistic missiles are a serious defiance to not only our country but also the international community."
The North issued photographs said to be of Mr Kim inspecting the launch.
North Korea's KCNA state media said on Saturday: "It can be said that Abe is the only one idiot in the world and the most stupid man ever known in history as he fails to distinguish a missile from a multiple launch rocket system while seeing the photo-accompanied report."
It added: "Abe may see what a real ballistic missile is in the not distant future and under his nose. Abe is none other than a perfect imbecile and a political dwarf."
Where are we with talks?
Negotiations between North Korea and the US remain stalled since the collapse of February's summit in Hanoi.
Mr Trump and Mr Kim did meet again in the Demilitarized Zone that divides the Koreas in June and agreed to restart working-level talks.
They began in October, but failed to make any progress.
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