WASHINGTON—The Federal Communications Commission on Thursday adopted a rule giving telephone companies wider latitude in preventing robocalls and spam text messages to consumers on both landline and wireless phones.

Automated calls and spam texts remain a major annoyance for consumers, agency officials say, despite the creation of a national “Do Not Call List” by the FCC and the Federal Trade Commission. FCC Chairman Tom Wheeler said the agency receives more complaints about robocalls than about any other issue, fielding...