TOKYO—A remark by Vice President Joe Biden that the U.S. “wrote Japan’s constitution to say they could not be a nuclear power” is getting attention in Japan, where possible changes to its pacifist constitution are on the agenda.

Mr. Biden made the remark in a speech in Pennsylvania on Monday, during which he criticized Donald Trump, the Republican presidential nominee. Mr. Trump has suggested in interviews that if Japan refuses to pay more to support the U.S. military presence in East Asia, it might have to defend itself without...