HONG KONG — For more than half a century, Asia has benefited from an American commitment to free trade and bilateral alliances in the region. But the election of Donald J. Trump left both statesmen and citizens here asking whether that commitment — and the prolonged era of peace and prosperity that has accompanied it — might be coming to an end.

Even before he began his campaign for the presidency, Mr. Trump was known as a critic of policies that opened markets in the United States to lower-cost Asian goods and helped fuel the dizzying growth that transformed Japan, gave rise to the dynamic economies known as the four Asian tigers, and propelled China’s emergence as a global power.

But during his campaign, Mr. Trump made a drumbeat of his criticism, accusing several nations here of currency manipulation and threatening a 45 percent tax on Chinese imports. He also added a geopolitical dimension by suggesting the United States might withdraw its security guarantees to Japan and South Korea unless they agreed to pay more of the cost of defending them.

Now, the question in Asia is how much of that rhetoric Mr. Trump will act on.

The Chinese president, Xi Jinping, sent a conciliatory message of congratulations to Mr. Trump that made no mention of Mr. Trump’s promises to confront Beijing on trade and currency issues. “Developing long-term healthy and stable Sino-U.S. relations is in the fundamental interests of the peoples of both countries and is also the shared hope of the international community,” Mr. Xi said.