TOKYO—Japan postponed an official visit by a United Nations representative for free speech, the latest move to prompt questions about the government’s commitment to freedom of expression.

David Kaye, U.N. special rapporteur for freedom of expression, said he planned to discuss a controversial state secrets act passed by Prime Minister Shinzo Abe’s government in 2013, among other issues. The law toughens penalties for leaking intelligence, but critics say it could be used to intimidate the press as wells as government whistleblowers...