Altice USA, the fourth-largest U.S. cable operator, said it plans to convert its entire network into an ultrafast fiber-to-the-home network capable of 10 gigabits-per-second speeds within the next five years, a plan that takes aim at the company’s fierce rival, Verizon Communications Inc.’s Fios.

With a 1 gbps connection, a customer can download a high-definition movie in about 30 to 40 seconds. A 10 gbps connection would download the movie in just a few seconds. With a 50 megabits-per-second Verizon Fios connection, it would...