By Nicole Gaouette and Elise Labott

Washington (CNN)--Secretary of State John Kerry and his Chinese counterpart locked horns over a range of issues Tuesday, agreeing to disagree in a State Department press conference over South Korean missile defense and the contested South China Sea.

But Kerry also pointed to an area of progress: nearing agreement on a U.N. Security Council resolution on North Korea's nuclear test. Even so, he defended the possible U.S. deployment of advanced THAAD missiles to defend South Korea and called for de-escalation as China and its neighbors compete over nearby islands.

The back-and-forth between Kerry and Chinese Foreign Minister Wang Yi reflected the awkward dance of competition and cooperation between the United States and China. The world's two largest economies work together on issues like Iran's nuclear program, the civil war in Syria and climate change, even as they clash over cybertheft, North Korea's nuclear ambitions, missile defense in South Korea and how to handle contested stretches of ocean in Asia.

And it came as U.S. officials confirmed that China has deployed fighter jets, J--11s and J-7s, to the disputed Woody Island. U.S. officials told CNN it was not the first time China has placed fighter jets on the island after expanding the runway there in 2014.

"We can cooperate in areas where our interests and values are aligned despite the fact that we have differences," Kerry said, while Wang said the United States and China "should make the cake or the pie of our common interests bigger."

That said, Wang acknowledged that theirs is "a complicated and diversified relationship."

Wang's visit also comes a week after an international outcry over newly discovered Chinese missile installations on Woody Island on the Paracel island chain in the contested South China Sea. The United States and Japan raised concerns about China's deployment of HQ-9 air defense missiles, with Vietnam going to the United Nations secretary general to complain that China had violated its national sovereignty.

Taiwan also claims the Paracels, one of two disputed Asian island chains that China says it owns.

The competing countries are vying for rich fishing resources and potential undersea oil deposits in an area that sees about $5 billion worth of annual shipping trade. The United States has consistently raised concerns that any restrictions on freedom of movement could affect the global economy and that militarization of the region makes it a potential powder keg in which an isolated incident could spiral into a geopolitical clash.

Wang insisted Tuesday that China and Southeast Asian nations could maintain calm in the region on their own. "And there have not been any problems with freedom of navigation in the South China Sea," he added.

The Asia Maritime Transparency Initiative, a project by Washington's Center for Strategic and International Studies that tracks East Asian maritime security, has said HQ-9 deployment on Woody Island "does not alter the military balance in the South China Sea."

It warned, however, that new Chinese radar facilities being developed in the Spratlys, the other disputed island chain, "could significantly change the operational landscape in the South China Sea" and speak to "a long-term anti-access strategy by China — one that would see it establish effective control over the sea and airspace throughout the South China Sea."

On Tuesday, Kerry addressed the radar installation, saying, "We have been very clear that it is important for all of the nations -- China, the Philippines, Vietnam, others -- not to engage in any unilateral steps of reclamation, of building, of militarization."

"The fact is there have been steps by China, by Vietnam, by others that have unfortunately created an escalatory cycle," Kerry added. "What we're trying to do is break that."

China has denied militarizing the islands, but Kerry's comments drew a sharp response from Wang. He didn't name the United States but referred to American flights and ship movements near the contested islands. "I hope friends in the media will not only see the radar, but, more importantly perhaps, that every day the advanced armaments and equipment emerging in the South China Sea," Wang said, "including the strategic bombers, the missile destroyers. Why people have chosen to disregard or ignore them?"

Kerry said the United States and China are close to agreeing on a United Nations resolution on North Korea's nuclear tests, the most recent of which took place last month. China is the isolated country's major protector and had objected to recent U.S. sanctions against Pyongyang, saying they would "further complicate things."

On Tuesday, Kerry said the two sides "agree completely that these actions merit an appropriate response through the United Nations Security Council" and that it would be coming soon.

And he was unapologetic about U.S. consideration of a missile defense system for South Korea. "Russia and China have obviously expressed concerns about THAAD," Kerry said. "We have made it very clear we are not hungry or anxious or looking for an opportunity to deploy THAAD."

The only reason the United States was considering THAAD, Kerry said, was because of North Korea's threats and its nuclear program.

"THAAD Is a purely defensive weapon, it is purely capable of shooting down a ballistic missile it intercepts and it is there for the protection of the United States," Kerry said. "If we can get to denuclearization, there's no need to deploy THAAD."

CNN's Barbara Starr, Pamela Boykoff and Steven Jiang contributed to this report.