This isn't true. Taking off anything vertically requires their thrust-to-weight ratios be higher than 1.0 (otherwise try as it might, it's going to sit on the ground).
Modern, latest-generation fighter aircraft have a TWR somewhere between 0.75 and 1.15. If you pointed them upwards, they will barely be able to lift off the ground (if their TWR is above 1.0 at all), much less gain enough speed to generate lift with their wings.
Taking off horizontally allows you to get lift "for free" with the wings. Accelerating horizontally means you're not fighting gravity the whole time. Lifting off vertically, assuming the aircraft has enough TWR to make it realistic, is a terrible waste of fuel. Modern fighters only have a combat radius of ~300-350mi before they have to turn around and go home. The more fuel that is expended taking off or landing, the less actual flight (and fight) time they have.
Catapult launches also allows you to save fuel on the aircraft by "borrowing" the necessary energy from the launcher. Ditto arrested-hook landings that allow you to borrow stopping force from the aircraft carrier - otherwise you'd have to carry enough fuel to exert this force yourself.
It's certainly possible to design craft that can take off and land vertically - the physics are relatively simple. The problem is making them practical - the more fuel/weight an aircraft spends just taking off and landing, the less weight can be dedicated to useful tasks, like missiles, bombs, and flying far.
Taking off horizontally allows you to get lift "for free" with the wings.
Well, more specifically, you are using the atmosphere as reaction mass. Wings direct air down. It's not free lift, it's free reaction mass.
I'm not saying that you should eliminate catapult launches. But a ship large enough to have catapults can still be a whole lot smaller than a modern US Navy aircraft carrier. Any aircraft that can land vertically will also be able to take off vertically. (Though with less efficiency when fully loaded, if at all.)
There's a huge amount to be gained by being able to stage aircraft from much smaller ships. How about building submarines a bit larger than the old Soviet Typhoons, and having aircraft on those? Short of that, how about the ability to build squadrons of multiple redundant cruiser-sized carriers?
Perhaps such schemes would be more valuable potential opponents of the US, than they would be to the US.
Yes, on the whole, the system will be less fuel efficient than horizontal take off and landing.
These are called amphibious assault ships like the America and Tarawa classes. They can serve helicopters and V/STOL fixed-wing aircraft. No catapults.
http://en.wikipedia.org/wiki/America-class_amphibious_assaul...
http://en.wikipedia.org/wiki/File:USS_Saipan_LHA-2_amphibiou...
This heralds the end of the aircraft carrier (a good thing, since they're really quite obsolete conceptually).
As to why it's useful for combat aircraft to be larger. First radar cross sections have little to do with the overall size of the craft it's mostly a question of shape an material assuming everything is staying within the same order of magnitude. Second while strait up thrust/weight is useful, top speed, range, and munitions capacity are generally more important because if your fast enough you can bank at over 10g's which will cause you to pass out long before the aircraft is in trouble. So while maned aircraft are still limited by the pilots all of the rest of those tend to improve as you scale up aircraft size.
PS: In many ways Cost and the desire for redundancy do limit the size of both aircraft carriers and combat aircraft.
A more direct comparison: A fully loaded standard F-35 has a thrust to weight ratio of 0.87 (that is with the afterburners on). In comparison the short take of and landing version of the F-35 loses 43% payload of its and 25% of it's fuel capacity (and thus range) [1].
1 - http://en.wikipedia.org/wiki/Lockheed_Martin_F-35_Lightning_...
I wasn't thinking of abandoning horizontal flight entirely. Being able to land a drone aircraft after it has spent its munitions and fuel should be handy.