New York state launched an investigation into Apple Inc.’s response to a bug in its FaceTime video-chat system that allowed callers to eavesdrop on others using the technology giant’s devices, deepening the scrutiny of a security setback that has undermined the company’s privacy position.

The investigation, announced Wednesday by New York Gov. Andrew Cuomo and Attorney General Letitia James, will focus on Apple’s failure to warn consumers about the glitch and its slow response to the issue, according to a statement from the...