Because an aircraft carrier can destroy a battleship. Carriers began destroying battleships in WWII. Prior to WWII, the battleships ruled the seas; after 10 December 1941, air
…power ruled the seas (carriers=airpower). Note: British battleship HMS Prince of Wales and the British battle cruiser HMS Repulse were sunk by Japanese aircraft on 10 Dec '41. .
Another reason that they don't have battleships is because there is almost no need for them. NGFS (Naval GunFire Support) is practically non-existent nowadays, that role is taken over by smaller gunned cruisers with 5-inch guns, and stealth and conventional bombers. There is no need for 16-inch guns, with guided and rocket assisted munitions in development for the 5-inch guns, and the bombers can be fitted with nuclear weaponry, something guns can't do. But the prime reason is because the battleships were aging. By the time the Iowas were recommissioned for Reagan's "600 ship Navy" program during the 1980's, the battleships were at least 40 years old, and some had been in reserve for 20+ years. By the time the last (USS Missouri) was decommissioned for the last time, the ship was at least 40 years old. Most of the ships now, with the exception of the USS Iowa, cannot be recommissioned, and are Museum ships. .
Actually, to counter the above, there is a severe need for Naval Gunfire Support ships. The current 5"/54 cal standard naval gun on all US destroyers and cruisers is significantly lacking in utility for effective NGFS role usage. This has been recognized for over a decade now, and the US Marine Corps has bitterly complained about the Navy's over-reliance on missiles and aircraft. There have been several experiments around a new 8" or 155m Naval gun, for use on the DD21 (Zumwalt) destroyers; the likely final winner is the Advanced Gun System from BAE. However, as it is expect that only 2 Zumwalt Destroyers will be ordered, this is still significantly below what the Marines (and Army) would like for real NGFS roles. Both branches were bitterly sorry to see the Iowa-class decommissioned, since these ships are arguably the best NGFS ever created. The real reason the Iowa class battleships (the last surviving operational battleships in the world, by almost 40 years) was cost. Building new battleships is massively expensive (even more than a Supercarrier), so no new ones have been built since mid-WW2. The reason the Iowa-class was finally decommissioned was that, as a 1940s-era ship, it was designed with a requirement for huge crew, over 2700 men, and lacked significant automation. Automating the Iowas to reduce the manpower requirement was deemed impossible, and the cost to run a 2700-man ship that wasn't an aircraft carrier was prohibitive (for comparison, a Ticonderoga-class Aegis cruiser has under 400 crew, and a Nimitz supercarrier has 3,200 crew + 2500 aircrew). It all boiled down to bang-for-buck. The Navy (which was unenthusiastic of the NGFS role), saw better utility in the newer Aegis cruisers and destroyers, and more firepower in a Supercarrier, so the battleships were phased out to make room for more cost-effective weapon systems. ( Full Answer )