Why does the Navy Enlisted wear their rank only on the left sleeve?

Best answer from:
https://bluejacket.com/naval_uniform_b.htm

Until 1949 rating badges were worn on either the left or right sleeve. From 1833 to 1885 Petty Officers of the Line (Seaman Branch) wore the Petty Officer Device on the right sleeve, and all others on the left sleeve except officers' stewards. Petty Officers of the Line (Seaman Branch) included deck ratings of boatswains mate, signalman and quartermaster, and ordnance ratings of turret captain, gunners mate, fire controlman, mineman and torpedoman.
From 1885 to 1913 the rating badge was worn according to the watch. The port section wore their badges on the left arm. From 1913 to 1949 the rating badge was again worn on the right by the Seaman Branch and the left for all others.

The eagle on the petty officer rating badge is derived from the Napoleonic eagle. This eagle was usually embroidered facing left. Why the Napoleonic eagle faces left is unknown. In 1941, the Navy changed the eagle's facing direction to follow the heraldic rules which face right toward the wearer's sword arm. This rule continues to apply and the eagle now faces to the front or the wearer's right. Bluejacket slang for the eagle is "crow."

and...tradition

While there is some interesting speculations, and other reasoning I would go with... Because we can and we don't think we need to over advertise just how bad azz we are. LOL ;-)

If you read between the lines, 1949 was just before the Korean War and the Navy was still in a downsizing mode from WW2. So what do they do? They fuss with things because they can. So "uniform" meant everyone would be "uniform". It likely wound up on the left side so you could still figure out what rank was saluting. All speculation, but this is the type of stuff that gets tossed around the table. BTW, I started out with the wool crackerjacks. The front trouser flap had 13 buttons for the original number of states (folklore). The uniform had inside creases because when you iron the wrinkles out of wool, the iron leaves a sheen where it makes contact. You turn the uniform inside out to put the sheen inside, thus preserving the finish on the outside. The uniforms today look more frumpy but are easier to maintain.