WASHINGTON — President Trump’s national security adviser, John R. Bolton, fleshed out his boss’s “America First” foreign policy on Monday, disparaging an international organization vilified by conservatives and reaffirming a pro-Israel tilt by announcing the closing of the Palestine Liberation Organization’s office in Washington. But on one of Mr. Trump’s signature projects, nuclear diplomacy with North Korea, he parted company.

On the same day that the White House welcomed a letter to Mr. Trump from Kim Jong-un, proposing another meeting of the two leaders, Mr. Bolton struck a markedly less optimistic tone, expressing frustration that Mr. Kim had not yet begun to fulfill his promise to rid North Korea of its nuclear weapons.

“The possibility of another meeting between the two presidents obviously exists,” Mr. Bolton said, “but President Trump can’t make the North Koreans walk through the door he’s holding open. They’re the ones that have to take the steps to denuclearize, and that’s what we’re waiting for.”

Mr. Bolton’s remarks came after a speech to the Federalist Society — his first major public appearance since taking his job in April — in which he threatened the International Criminal Court with sanctions if it investigated American troops in Afghanistan. His announcement of the closing of the P.L.O. office reinforces how far the White House has moved from the role of a broker in the region.