Over the past three months, nations across the world tried to avoid new tariffs that would punish their economies by giving President Trump something he might want.

Indonesia offered to buy $34 billion more in U.S. crops and fuels. Thailand proposed lowering many of its own trade barriers, and buying more U.S.-made planes. Japan was ready to buy more liquefied natural gas over the next two decades.

But as Mr. Trump’s self-imposed July 9 deadline approached, those entreaties made little difference. The 14 letters he posted online on Monday, mostly aimed at countries in Asia, largely matched the rates set in April, before he backed off and gave dozens of countries 90 days to negotiate agreements that would satisfy the White House’s demand for more balanced trade.

“We have had years to discuss our Trading Relationship with Thailand, and have concluded that we must move away from these long-term, and very persistent, Trade Deficits engendered by Thailand’s Tariff, and Non-Tariff, Policies and Trade Barriers,” Mr. Trump wrote, swapping out only each country’s name in otherwise virtually identical missives.

That fresh volley has left countries large and small, nearly all of them longstanding allies of the United States, with profound questions about how to move forward with the world’s largest consumer economy when negotiations over trade conflicts are labored and deadlines are extended without warning.

“Many in Asia are going to ask, ‘Is this how the U.S. treats its friends?’” said Manu Bhaskaran, chief executive of Centennial Asia Advisors, a research firm. “Will there be permanent damage to American standing and interests in Asia and elsewhere through these crude threats and unpleasant language?”