Moon's comments are similar to those made by President Donald Trump in late April.
Moon, who was sworn in last week, warned that the North's nuclear and rocket programmes were "advancing rapidly", days after Pyongyang launched what appeared to be its longest-range missile yet.
"This year marks the 25th anniversary of the establishment of diplomatic ties between South Korea and China, and we need to respect the things that we have achieved", he said.
"You either support North Korea or you support us, because that's what it comes down to", Haley said at a news conference.
Its latest ballistic missile launch, in defiance of United Nations 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.
Seoul and Washington have argued that the missile system is aimed at North Korean aggression, while China sees it as a threat to its own security.
North Korea was also under scrutiny by investigators looking for the source of a major cyberattack affecting more than 150 countries, but the European police agency said it was still too early to determine who caused the chaos.
Mr Trump has warned that a "major, major conflict" with North Korea was possible, and sent the Carl Vinson aircraft carrier strike group to Korean waters for drills with South Korea and Japan. Previous remarks by Trump suggest that he will strongly want the issue of renegotiating the South Korea-US Free Trade Agreement to be added to the agenda as well.
When asked what the USA would do if it was confirmed that North Korea had been behind the major ransomware attack plaguing computer systems across the globe, Haley said: "Sadly, it wouldn't be a surprise".
"President Trump talked mainly about big pictures, and emphasized that we can achieve an outcome through the strong unity of the alliance and worldwide cooperation", Hong said.More news: Steelers release veteran long snapper Greg Warren
The U.S., which has 28,500 troops in South Korea to guard against the North Korean threat, has called on China to do more to rein in its neighbor.
"(President Trump) talked about sanctions on North Korea and cooperative relations with China.
North Korea has been developing a long-range missile capable of striking the mainland United States mounted with a nuclear warhead.
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.
Despite the push for a tougher stance, she held out the prospect of direct talks with North Korea, saying "we are willing to talk but not until we see a total stop of the nuclear process and of any test there".
The aide, Matthew Pottinger, Asia director on the National Security Council, met with Chung Eui-yong, an adviser to the new South Korean president, and other foreign-policy aides in Seoul, the capital.
Hong said South Korea had not yet received official word from the U.S. on whether Seoul should pay for an anti-missile United States radar system that has been deployed outside Seoul.
But there are questions over whether Pyongyang can miniaturise a nuclear weapon sufficiently to fit it onto a missile nose cone, or has mastered the re-entry technology needed to ensure it survives returning into Earth's atmosphere.
In December of a year ago, satellite photographs captured the image of a North Korean underground missile silo that resembled a known silo in Iran.