The South Korean navy said the drill is aimed at "deterring North Korea's provocations and displaying the firm alliance between the United States and South Korea".
The meeting of the top United Nations body on Friday laid bare major differences among key powers over the way to address the North Korea crisis.
"The nuclear force of the DPRK just serves as a treasured sword of justice and creditable war deterrent to protect the sovereignty and dignity of the country and global peace from the United States threat to ignite a nuclear war", Kim said.
The THAAD is meant to protect targets in the area around Korea from North Korean missiles, though its effectiveness in actual situations is seen by most experts as extremely limited, and the it would be of little use if the U.S. did launch a full-scale war against North Korea, as most retaliation would be artillery, not missiles.
South Korea's defence ministry said it suspected Saturday's missile test had failed after a brief flight, while the U.S. military's Pacific Command confirmed the rocket did not leave North Korean territory.
The U.S. secretary of state also called on China, Pyongyang's main ally, to assist in efforts to keep North Korea's weapons program in check, saying, "We look forward to further actions to what China has already done".
The United States, Russia and China took part in six-party talks on North Korea's denuclearisation from 2003 to 2009, along with Japan, South Korea and Pyongyang.
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Pressure from the White House and support from conservatives could put intense pressure on moderates to vote for the bill. Rand Paul (R-Ky.), one of the lead Senate agitators against the House health care push, said Wednesday.
Japan condemned the launch as unacceptable and authorities stopped some train services in Japan as a precaution, in case the missile had been fired at Japan, a transit system spokesman said.
Pressed on whether that meant military action, Trump said, "You'll soon find out".
The latest test came as United States officials pivoted from a hard line to diplomacy at the United Nations in an effort to address what may be Washington's most pressing foreign policy challenge.
"All options for responding to future provocations must remain on the table", Tillerson said.
Meanwhile, a US aircraft carrier and supporting warships arrived on the Korean peninsula and began exercises with the South Korean navy late on Saturday.
Norwegian foreign minister Borge Brende said on Saturday on Twitter that "new missile test violates SC (Security Council) resolution".
China has threatened North Korea with sanctions if the regime conducts a nuclear test, Tillerson told Fox News on Thursday.
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They also looked back at their shared past, and Kris recalled how her makeup used to go missing mysteriously. And you have a better chance - although the Republican Party has very disappointed me over the years.
"They seem to be pretty intent on developing the capability, both missile technology as well as nuclear technology".
Moon has advocated a more moderate policy on the North and been critical of the deployment of an advanced US missile defense system in the South meant to counter North Korea's missile threat, which China also strongly objects to.
But condemnations and sanctions resolutions since 2006, when North Korea conducted its first nuclear test, have done little to impede its push for ballistic missiles and nuclear arms.
In a rare interview with CNN last week, a North Korean official said the country's nuclear tests would "never stop".
South Korea's navy has begun joint exercises with U.S. ships, including the supercarrier USS Carl Vinson, in a show of force amid the growing threat posed by North Korea's nuclear and missile programmes.
The statement, which came in response to CNN's questions about the latest launch, did not acknowledge Saturday's missile test.
The United States is ready to impose sanctions on third countries where companies or individuals are found to have helped North Korea's military programmes, he said.
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He's trying to be somebody-like a lot of these young guys are trying to be somebody that they're not-and he's trying to talk crap. While he did experience a two-fight skid in 2014 and 2015, he has gotten himself back on track with a four-fight win streak.
The US says that's not going to happen, citing the need to be prepared.