What is the date commemorating the birthday of the US Navy?

The date commemorating the birthday of the US Navy is indeed 13 October 1775. This date marks the establishment of the Continental Navy, which was created to engage British combatants during the American Revolutionary War. The formation of the Continental Navy represented the United States' initial step toward building a distinct naval force, which eventually evolved into today's US Navy.

Other dates listed do not specifically reflect the official establishment of the Navy. For instance, January 1, 1776, does not have a direct association with naval history in the context of the Continental Navy's formation. July 4, 1776, is celebrated as Independence Day, marking the adoption of the Declaration of Independence, but it does not pertain to the founding of the Navy. Similarly, April 14, 1775, predates the establishment of the Continental Navy and does not connect with any recognized naval events. Thus, 13 October 1775 is the acknowledged date that officially celebrates the inception of the naval forces that would later become the US Navy.