It is my understanding that the biggest problem with the naval version is that it has only one engine. The F-4 and F/A-18 were both twin engine planes and if one engine failed or was damaged, the plane could still fly and the pilot could land on the aircraft carrier and save the plane. With the F-35C, if the engine is damaged far from the carrier, the pilot will probably have to eject and be rescued. In that case, even if cost is no object there is the inconvenience of going to rescue the pilot and being short one airframe. I also remember reading somewhere that the repair bays on an aircraft carrier can swap an engine in an F/A-18 in just a few hours.

Well, the Navy's been there fairly recently, the single engined A-4 Skyhawk, the F-8 Crusader, and the A-7 Corsair which was based on the latter all saw heavy service in Vietnam. And per Wikipedia, the latter replaced the A-1 Skyraider (to the extent a jet like that could) and the F-100.

But for the obvious reasons they vastly prefer two engined planes, plus I've read the F-35's single engine is so big their current logistics system will have trouble getting new ones to a fleet at see, I assume this means it can't fit in a C-2 or maybe V-22.