Final answer:
The CN III (oculomotor) nerve innervates the inferior rectus muscle, which is involved in downward gaze and inward rotation of the eye.
Step-by-step explanation:
The cranial nerve that innervates the inferior rectus muscle is the CN III (oculomotor) nerve. This nerve is responsible for controlling most of the eye's movements, including constriction of the pupil and maintaining an open eyelid. The inferior rectus muscle is one of the extraocular muscles that enables eye movements such as downward gaze and helps in contributing to inward rotation of the eye.