Actually, no, it can't. The planes have to allow each other to attack, which lowers the amount of ordinance they can deliver compared to a battleship. It would take a battleship between 9 and 20 minutes, depending on how many barrels were firing, to deliver 210 tons of ordinance. A Nimitz class carrier would take 12 hours.
With advances in targeting, munitions delivered from the air are more accurate than in the past, but many of the same advances can and have been applied to weapons systems on battleships.
In addition, as North pointed out, the planes can't work under all weather conditions. The ships can. The proposed 5" round that North mentions cannot be as effective against reinforced structures as the 16" round, which in its AP form is designed to cut through 20" of reinforced concrete (IIRC -- am I right on this, Junior?). The issue is not as cut and dried as some posters here would portray it.
For instance, I would say both posted examples of battleships being sunk in the past are not applicable to today. The Bismark was hunted down by combined forces including battleships, an unfortunate battle cruiser, cruisers, and an aircraft carrier. They all combined to kill it, which only goes to show that any ship can be overwhelmed. Yamato (and it's sister ship, Musashi), were sunk by planes bearing weapons designed to kill a battleship.
Care to name the modern weapon designed to pierce the kind of armor carried by battleships? There aren't any, because a weapon that would pierce armor would cut right through a modern destroyer without exploding. The weapons that crippled all the examples of destroyers and frigates posted here would bounce off a battleship's armor. And attempting to place an armor-piercing head on a missle without completely redesigning it would -- to use a technical term -- throw it out of whack.
On the other hand, Junior (a Navy man himself), makes a good point about manning requirements. But that same problem, including the need for escorts, also applies to the Navy's carriers. A battleship's compliment is in the range of 1,500 to 2,000 men, with the lower number more applicable to the last configurations of the Iowa class. A modern carrier's compliment is in the area of 5,000.
I believe many of the technological problems involved -- the sixty year old bits of the battleships -- can be largely overcome, but that will also require money.
So, as I said, I don't think it's cut and dried. It's worthy of additional study.
I disagree. North wants to preserve battleships for amphbious assaults of a hostile beach, a scenario that has not been seen for over 50 years. It makes no sense to sink billions of dollars into an asset with limited flexibility and limited use. Carriers can strike targets hundreds of miles from the shore, something a battleship cannot. Carriers can strike multiple targets hundreds of miles apart, something that a battleship cannot. Carrier aircraft, contrary to what North says, can operate in virtually all weather conditions. Battleships would have been worthless in the most recent combat in Iraq and Afganistan, while carriers carriers much of the load. Carriers provide a flexibility that battleships do not, and represent a much better investment in ever-scarcer defense dollars.
You discount nuclear-tipped sea-skimming missiles. They exist. For one example:
http://www.fas.org/man/dod-101/sys/missile/row/moskit.htm
And, you don't have to sink a ship, to take it out of the fight. A "mission kill" happens at a much lower threshold of damage.