Moon made his comments hours after South Korea, which hosts 28,500 USA troops, said it wanted to reopen a channel of dialogue with North Korea as Moon seeks a two-track policy, involving sanctions and talks, to try to rein in its neighbour.
Meanwhile, South Korean President Moon Jae-in's envoy for China, Lee Hae-chan, had not met Chinese President Xi Jinping as of 0540 GMT, foreign ministry spokesman Cho June-hyuck said in a regular media briefing.
Haley said the U.S. and China have been working on "a unified plan" on how to approach North Korea that would include stronger implementation of existing sanctions and tougher new sanctions.
Nuclear-tipped ballistic missiles in the hands of the "volatile" Kim Jong Un are "a recipe for disaster", the head of the U.S. Pacific Command (PACOM) said on Tuesday.
The two countries are technically still at war because their 1950-53 conflict ended in a truce, not a peace treaty.
Its latest ballistic missile launch, in defiance of UN Security Council resolutions, was on Sunday which it said was a test of its capability to carry a "large-size heavy nuclear warhead", drawing Security Council condemnation.
Yoon said Chung and Pottinger in their meeting reaffirmed that Seoul and Washington share a common goal in the "complete discarding" of North Korean nuclear weapons and will pursue "all methods, including sanctions and dialogue" to reach the goal. The demand runs counter to a deployment agreement in which the US agreed to pay for the system if the South agreed to host it and provide land.
"This is very different to dialogue, or talks, or anything approaching what might come down the line - but at its most basic it is a sign that the South's new government is trying to be more reconciliatory towards the North despite Sunday's successful test on Sunday and Washington's talks of sanctions at the UN", Thomas said.
The United States said the missile landed close to Russian territory, but Moscow later said it fell in the ocean about 500km away and posed no threat.
The US may push to impose sanctions on countries supporting North Korea, US Ambassador to the UN Nikki Haley said Tuesday.
Nikki Haley spoke ahead of a closed meeting of the U.N. Security Council called by the US and council member Japan, along with South Korea, to discuss Pyongyang's latest illicit ballistic missile launch.
Trump warned this month that a "major, major conflict" with North Korea was possible, and in a show of force, sent the Carl Vinson aircraft carrier strike group to Korean waters to conduct drills with South Korea and Japan.