Shinzo Abe, Japan's prime minister, called on the international community to band together after the projectile flew over the northern Japanese island of Hokkaido before falling into the Pacific Ocean on Friday.
"The latest sanctions need to be imposed in full," he said. "North Korea needs to be made to understand that if it continues on this path, there is no bright future for it."
Al Jazeera's Craig Leeson, reporting from Tokyo, said that the Japanese government had expected the missile launch given North Korea's strong rhetoric on Thursday, threatening to sink the country with a nuclear weapon.
"Japan expected this kind of retaliation because of the support it provided to the US with those strengthened sanctions that were pushed through in the UN Security Council on Tuesday."
For his part, South Korea's President Moon Jae-in ordered his military to conduct a live-fire ballistic missile drill.
Al Jazeera's reporter Andrew Thomas, reporting from Seoul, said: "They are demonstrating there that they have the capacity to fire at a launch site immediately or preemptively. They are watching and are ready to respond."
Moon said having a dialogue with North Korea is "impossible", and the country should face stronger sanctions and pressure from the international community.
"We should be equipped with protecting ourselves from North Korea's growing nuclear and missile threats and to eliminate those threats by quickly punishing them in cases of provocation against us," he said
The White House said US President Donald Trump had been briefed on the latest launch.
Trump wants China to do more to rein in North Korea. China, in turn, favours an international response to the problem.
James Mattis, the Pentagon chief, said the launch on Friday "put millions of Japanese into duck and cover", although residents in northern Japan appeared calm and went about their business as normal after the second such launch in less than a month.
Rex Tillerson, US secretary of state, called for "new measures" against Pyongyang and said the "continued provocations only deepen North Korea's diplomatic and economic isolation".
"China and Russia must indicate their intolerance for these reckless missile launches by taking direct actions of their own," he said.
Taiwan calls meeting
Taiwan convened a national security meeting, amid fears of a deterioration in regional security after North Korea's latest missile launch.
"Our government condemns North Korea's repeated threats of force damaging regional safety and stability," President Tsai Ing-wen's office said in a statement.
Tsai also called on security and diplomatic agencies to carefully monitor developments and keep close contact with friendly nations to jointly maintain stability and order in the region, according to the statement.
The UN Security Council has called for an emergency meeting on Friday.
Al Jazeera's Diplomatic Editor James Bays, reporting from the UN headquarters in New York, said North Korea's missile launch is being seen as an "act of defiance" by the Security Council.
He said the meeting would discuss the situation and the "provocative timing" of the missile launch, which came just 72 hours after the last Security Council meeting in which additional sanctions were imposed on North Korea.
Friday's missile launch also came just days before world leaders meet in New York for the UN General Assembly session, which will be attended by a high-level North Korean delegation led by the country's foreign minister.
While the North Korean delegation typically does not hold meetings with representatives from other countries, Al Jazeera's Bays said that in the past the North Korean foreign minister has met the UN secretary-general.
"It is possible that there could be a very important meeting with the new secretary-general, Antonio Guterres," he said.