Pentagon officials said on Friday that they were trying to determine if the seizure of the underwater drone had anything to do with Mr. Trump’s comments.

At the White House on Friday, President Obama was asked about the issue during a news conference, and he made clear that he viewed the question of Taiwan as especially sensitive. While the president refrained from directly criticizing Mr. Trump, he warned his successor to carefully consider his actions and any new policy, lest he ignite what could be a significant response from Beijing.

“I think all of our foreign policy should be subject to fresh eyes,” Mr. Obama said. But he added: “For China, the issue of Taiwan is as important as anything on their docket. The idea of a One China is at the heart of their conception of a nation.”

“And so if you are going to upend this understanding, you have to have thought through what the consequences are, because the Chinese will not treat that the way they’ll treat other issues,” he said, adding that the Chinese would not even treat it the way they treated issues around the South China Sea, “where we’ve had a lot of tensions.”

China experts said on Friday that it was unclear whether the seizure of the American drone was linked to anger in Beijing over Mr. Trump, or a continuation of years of tensions over competing claims in the South China Sea.

The Bowditch episode came after China signaled on Thursday that it had installed weapons on disputed islands in the South China Sea that it would use to repel threats. In describing the new weapons deployment, a Defense Ministry statement suggested that China was further watering down a pledge made by its president, Xi Jinping, to not militarize the islands.