Shareholders of Walt Disney Co. and 21st Century Fox approved the $71 billion deal between the two companies on Friday, clearing another hurdle for a deal that will rattle the media and entertainment landscape and could inspire a wave of similar tie-ups.

If the deal is completed, Disney will absorb Fox’s storied film and television studios, responsible for franchises like “Avatar” and “The Simpsons,” as well as stakes in the Sky PLC pay-television company and the Hulu video-streaming service.

Disney...