ALBANY, N.Y.—Gov. Andrew Cuomo’s administration said Wednesday it would prohibit hydraulic fracturing statewide, citing health concerns and calling the economic benefits to drilling in New York state limited.

The long-awaited decision seals off about 12 million acres of the Marcellus Shale, an underground rock formation with natural-gas reserves that have helped fuel an energy-production boom in Pennsylvania, West Virginia and Ohio. New York would be the first state with significant potential to become a major natural-gas...