Countries in Southeast Asia worried about the near collision, while both China and the United States want to keep up relations with those countries to counter the influence of the other power.
"I think there is every reason, every incentive for both [the] U.S. and China to at least create the appearance that they are both keen to stabilize the South China Sea," Koh said.
China cites historical maps to call about 90 percent of the sea its own, despite competing sovereignty claims from Brunei, Malaysia, the Philippines, Taiwan and Vietnam. It has angered the others, all militarily weaker, over the past decade by building infrastructure on disputed islets for bombers, missiles and radar systems.
The United States has no claim to the 3.5 million-square-kilometer sea but wants it open internationally for freedom of navigation. The U.S. Navy routinely sends vessels into the sea. It does annual joint naval drills with the Philippines, and in 2016 the U.S. government lifted a ban on selling certain weapons to Vietnam. It also uninvited China to the multi-country RIMPAC wartime exercises in June and July.
At the meetings Friday, Mattis "raised the importance of all maritime forces conducting their operations in a safe and professional manner in accordance with international standards and norms," the statement said. U.S. Secretary of State Mike Pompeo and Chinese State Councilor Yang Jiechi also attended.
U.S. President Donald Trump and Chinese President Xi Jinping will meet in Argentina during a summit of the G-20 major economies, and some of the talks last week were "centering" on that summit, a U.S. State Department statement said. Heads of state generally try to avoid upsetting each other before summits.
Xi and Trump are more likely to touch on their 10-month-old trade dispute than the maritime dispute, some believe. The U.S. government has approved import tariffs this year covering $250 billion in Chinese goods.
But the two leaders have the options of exchanging views and making upbeat pledges about the maritime dispute, Koh said.