New York City notified Verizon Communications Inc. Tuesday the company was in default of an agreement to build Fios cable connections across the city, escalating a fight that began last year.

The city alleged last summer that Verizon has failed to make its Fios network available to every home as required under the 2008 agreement. After more than a year of talks, city officials say Verizon still hasn’t made meaningful progress.

The...