Why was the P-51 Mustang not adopted by the U.S. Navy?
The biggest driver behind the use of the P-51 Mustang in naval operations was the need for B-29 escorts while bombing Japan. No naval fighter at the time had the range to accompany the B-29's so the P-51D was selected to determine if it could be a suitable carrier operator.

The test project was named "Project Seahorse", and one aircraft was selected and specially modified to cope with carrier landings. 


Navy Concept Image in Livery (Image Source)

The modifications included:


"Fin Fillet Extension" which enhanced low-speed operation.
Tail hook (for obvious reasons)
Catapult hook 
High pressure tires 
Higher pressure in the shock absorbers
Reinforced airframe


The modified P-51D started testing in September of 1944 under test pilot Robert Elder. The tests were carried out at Mustin Field in Phillidelphia on a modified runway fitted with a catapult and arresting cables. Throughout September 1944 to October 1944, 150 launches and recoveries were made on this modified runway. One big concern was that the aircraft would need to land around 90mph and the stall speed was 82mph, not much of a margin.

In late October 1944 the next stage of testing started, carrier landings/launches at sea. Lt. Elder made all the carrier landings at 85mph and was pleased with its handling characteristics, but there were issues:


Stall speed margin was extremely low, too low for safety
Rudder control at low speed and high angles of attack was inadequate
Landing attitude had to be very carefully controlled to avoid airframe damage
Go-arounds required slow throttle advancement. The extreme power of the Packard/Merlin engine meant that a high-power fast throttle advancement could put the aircraft into a roll or snap-roll. At low speeds this would prove a fatal mistake. 


Only 25 launches/recoveries were made in the suitability trials and Lt. Elder did not believe that the Mustang had a place in carrier operations.

It was a moot point anyway. By early 1945 Okinawa and Iwo Jima were won by the allied forces. Each island had airfields that were taken over and provided a close base of operations for aerial attacks on Japan. In August 1945 Japan announced its surrender and it was official September 2, 1945. The P-51 wasn't required to be used from a naval platform anymore.

The project wasn't forgotten though. North American presented the NAA-133 based on the P-51H, which was a carrier operations "Mustang". It didn't go any further than the design stage though. In 1947 a P-51H was acquired to test new catapult equipment. Either way the war was over by that time and we didn't have to put any more effort into modifying the aircraft for carrier operations.

Here is my main source