With it's airwing, a carrier can certainly protect itself better than a battleship depending on nothing more than CIWS. Battleships would require escorts as well.
Carriers cannot defend themselves adequately against submarines, again requiring escorts. A BB can screen a carrier from modern torpedoes, taking several hits which would severely hamper a carrier's ability to operate (not necessarily sink it -- it's not easy to sink carriers with torpedoes nowadays).
And who will screen the battleship, which has no ASW capabilities at all? The same assets that screen the carrier - destroyers with helicopters and our own submarines. In addition, the carrier will carry its own ASW helicopters and support centers to coordinate them. The battleship will not.
Carriers can engage in gunfire duels with much better effect than today's escorts armed with 5" or 76mm guns.
Assuming you meant battleships, the last surface gunfire dual on the high seas predates the last amphibious assault of a defended beach. If anything on the surface comes close enough to a carrier to shoot at it, then something went seriously wrong.
Battleships have almost the same range as a carrier air wing thanks to Tomahawk missiles.
You can say the same about destroyers, frigates, and submarines. All newer platforms, more flexible, and less expensive to run than battleships.
The ships still complement each other rather nicely, I think, although whether the Iowas will ever be reactivated again is an open question.
I'm sorry, but I believe battleships are a relic from several wars ago, and have no place in the modern navy.
I was an AO2 aboard the Enterprise, Midway and Eisenhower in my 8 years in the Navy. I worked on f-14's, A-6's and S-3's. Trust me, if there were enough planes, or missiles in the air, F-14's and CIWS would not have gotten them all. An F-14 is capable of attack six targets at a time, with Phoenix missiles, which have gone away for good.
And ASW is bitch. It's harder to defend against submarines that it is aircraft.
CIWS is only good out to 1.5 or so nautical miles.
Aegis is capable of tracking upwards of 200 targets at a time, but the typical Aegis-equipped ship can only attack 92 of them, not counting the time taken for reloading. Nor does it take into account that several ships might defend against the same target.
In the scenarios we trained for, it was not unusual to have the "enemy" launch enough ordinance to swamp those defenses.
The battleships were designed and deployed long ago as strategic platforms. Gun boat dipolats. The nuclear era gave us a new and powerful strategic tool. The era of gunboat dipolmacy is over. Smaller tactical vessles although vunerable, certainly accomplish the our Naval Maritime Mission's. Present and future.