Tillerson's comments at a regional security forum in Manila were the latest USA attempt to rein in Pyongyang's nuclear and missile programs after months of tough talk from U.S. President Donald Trump.
Cambodia has previously made clear it will not back any further nuclear development efforts by North Korea.
Mr Ri was avoided by Mr Tillerson, who skipped a group dinner attended by his North Korean counterpart.
China's foreign minister said Sunday that new U.N. Security Council sanctions on North Korea were the right response to a series of missile tests, but insisted dialogue was also vital to resolve the complex and sensitive issue now at a "critical juncture".
Wang said the two had an intensive conversation during which China urged North Korea to maintain calm.
Ri hasn't spoken publicly since arriving in the Philippines.
"Minister Kang strongly condemned the North's repeated provocations and emphasized the need for the worldwide community to be more united and faithfully carry out the U.N. Security Council's Resolution 2371", the ministry said in a press release.
North Korea has launched more than a dozen test missiles this year.
The US will hold the door open to North Korea if it halts a recent series of missile test launches, said Secretary of State Rex Tillerson. The US-sponsored resolution, which passed with unanimous support from all 15 council members, is going to hit Kim Jong-un where it counts, denying the country an estimated $1 billion in exports or about one-third of its total trade with the outside world. Tens of thousands of them return hard currency to the increasingly isolated regime.
Tillerson aide R.C. Hammond says that after a productive first day, Tillerson has taken time to prepare for Day 2.
Earlier, U.S. Secretary of State Rex Tillerson said new United Nations sanctions targeting North Korea are a "very good outcome".
Since North Korea's first nuclear test in 2006, the United Nations has imposed ever-tightening sanctions on the rogue regime to force it to halt its weapons programs. Very big financial impact!
"Owing to China's traditional economic ties with North Korea, it will mainly be China paying the price for implementing the resolution", the statement cited Wang as saying.
US Secretary of State Rex Tillerson said the new UN measures were a stern message to North Korean leader Kim Jong Un that the worldwide community is united in its commitment to see a Korean Peninsula without nuclear weapons. To the administration of USA president Donald Trump, who's been blaming Beijing for not doing enough to rein in Pyongyang, it seems to be saying, the ball is in your court.
Moon's office said that he and Trump had agreed to apply "the maximum pressure and sanction".
"It's not that easy but it is a direction we need to work together towards", Wang said of the six-country talks, which used to be hosted by China and were stopped in 2009.
"If North Korea continues to pursue its nuclear programme, which is extremely likely, then it'll just be another failure on the part of the entire global community", Zhang said.
The relationship between China and North Korea is not quite as special as it once was. "It is China", Wang, the Chinese foreign minister, said Sunday. Trump had been stressing how China could play a large role in containing North Korea.
Speaking to reporters after the council vote, Washington's ambassador to the UN Nikki Haley said "what's next is completely up to North Korea". "They never have and likely never will". He says the only way that would change is if another country joined in an American action against North Korea.
'I don't think that matters much, ' he added.