You can, but you'll be considered AWOL and a fugitive from then on. Trust me when I say that the military does not have a sense of humor when it comes to deserters or those who just leave.
The only ways of leaving are by Medical, Honorable, other than Honorable, or Bad Conduct Discharge. Military service contracts are extremely binding - before you sign, you'd better understand exactly that you're giving up your rights for however long your hitch is.
There is one way that some sailors have beaten the system, but it's with the assistance of a national government, and that was many years ago - it may not be the same today. Usually when ships visit Australia, they leave with 1 or more fewer crewmembers. Australia is a country that has a high female - male ratio - there aren't that many men, and the Australian government tries to attract capable individuals. But make no mistake - you'd still be a wanted fugitive in the U.S., and would never be able to return. There's always something that you miss that will get you caught.
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Yes - the U.S. Navy is considered the largest Navy in the world.
The British Royal Navy is far more advanced, modern and technologically sophisticated than the Indian Navy. The sailors of the Royal Navy are also more experienced.
Nothing - Department of the Navy is the formal title for the Pentagon division that encompasses the U.S. Navy and the U.S. Marine Corps. Navy Department is an informal term.
In 1914 the US had the third largest navy in the world. It ranked below the number one sized navy of Britain, and the second largest navy in 1914 was the German navy.
Of course, navy aircraft are military aircraft.