Mr. Trump’s aides did not respond to questions about whether the president-elect had used his own interpreter in the meeting with Mr. Abe and calls with other leaders, or whether he had relied on interpreters provided by them.

“The order that the calls are done in seems like a silly protocol issue. But it’s one that people really think about,” said Loren DeJonge Schulman, a senior fellow at the Center for a New American Security and a former top aide to the national security adviser, Susan E. Rice. “They are taking messages away from that, whether you think it’s important or not.”

Mr. Abe’s visit — Mr. Trump’s first with a foreign leader since the election — came on a day when the president-elect was steeped in foreign policy. He met with former Secretary of State Henry A. Kissinger to discuss “events and issues around the world,” ranging from Russia and China to Iran and the European Union, according to the transition team.

He met with Ron Dermer, Israel’s ambassador to the United States, who told reporters afterward that “Israel has no doubt that President-elect Trump is a true friend of Israel.” Mr. Dermer said he looked forward to working with “all of the members of the Trump administration, including Stephen K. Bannon,” Mr. Trump’s chief strategist, who has been accused of promoting anti-Semitism through his right-wing news media operation, Breitbart News.