You might look at the various US carrier raids through the Central Pacific of 1943 and 1944. The Fast Carrier force just demolished the large island based squadrons Japan had been counting on to defend the Marshall and Marianas islands.

The crowning achievement then was in February of 1944 on Truk Lagoon, the major forward IJN fleet base, Japan's Pearl Harbor in a way. After 3 days 250 Japanese aircraft were destroyed and about 80% of the shipping in the harbor and shore installations were useless. At one point Admiral Spruance even took a fast squadron of heavy ships including the battleships New Jersey and Missouri in close to finish off a few Japanese destroyers.

Alternatively you might look at the efforts by Japan to land reinforcement and retake Guadalcanal. While US carriers provided both close and distant cover they took a beating. And the cornerstone of the defense was Henderson Field and the motley CACTUS Air Force based there.

Hmm an interesting out-of-the-box question.

Generally the answer to your question is no. Ships without air cover proved to be quite vulnerable to air attack during World War Two. Even battleships, and in the few cases where battleships did challenge air power the ships generally came off distinctly worse. The key problem, of course, is that a battleship's guns only have a range of 10-15 miles. Whereas a WWII bomber could fly 200 miles, or more. If I put my airbases at least 20 miles from the coast then your battleships can't reach them and I can hit you with impunity.

However there is one example from the war where battleships did directly challenge airpower and almost won.

During the contest for the island of Guadalcanal in the Solomons Island chain, the US was trying to build up their forces on the island. Air cover was primarily provided by fighters and bombers flying out of Henderson Field. Japanse forces, and their supply convoys, on the island were subject to near constant bombardment. Japanese bombers flying from Rabaul had only limited effect on the American airbase because of the long range and the early warning system that the allies had in place.

But Guadalcanal is not a large island and the airbase is almost directly on the coast. The Japanese Navy made several bombardment runs including a shelling, by the battleships Kongo and Haruna, on October 14th 1942 which inflicted serious damage on the facilites and aircraft. Ground forces were unable to exploit this success though and another operation was planned for mid-November.

On the night of 12-13th November the Japanese battleships Hiei and Kirishima, and their escorts, on their way to shell Henderson field ran into an Allied force of heavy cruisrs. During the close-quarters night-time battle that followed both sides lost heavily and the Japanese commander elected to withdraw. Hiei had been critically damaged though and was sunk the next day by bombers flying from Henderson.

The following night Kirishima tried again but this time ran into the US battleships South Dakota and Washington which had just arrived having been ordered to the area by Admiral Halsey. Kirishima was sunk and the Japanese withdrew. This was the last time the Japanese battleships attempted to shell Henderson (and one of only two cases where battleships classed during the Pacific War).

Had the US Navy not been on hand to counter the Japanese battleships they might well have been able to inflict enough damage on the airbase to neutralize it and escape the following day. However, this is obviously the result a very special set of circumstances. Henderson field was isolated from other allied bases and was practically on the coast. US operations on Guadalcanal were limited by supply problems. With their own nearby bases the Japanese were able to close in to Hendersen during darkness which prevented the bombers there attacking during their approach. Though this did leave them vulnerable during their escape.

James Hornfischer, Neptune's Inferno

Please note that this is a very brief summary of the events around Guadalcanal during the summer and autumn of 1942. I have left out a great many details.