Could flywheel type gyros stablize a plane?
I actually have a lot of experience in the shipboard systems that you are talking about. 

First, the gyro's do not keep the boat upright, they counter the rolling effect of waves but they cannot completely remove them. Gyro's of all designs work on the principle of conservation of angular momentum. That means that a spinning object will tend to impart a force 90° opposite of the force applied to it to conserve angular momentum. 

In order for this to have a tangible effect on something of significant mass, the gyro itself (more specifically, the flywheel) must also have a significant mass. In shipboard systems this means that they usually use many (for an aircraft carrier, I would assume dozens even) of these devices that spin a flywheel weighing in the magnitude of tons. That is a lot of mass to be moving around!

So the first issue you have is to get the mass spinning fast enough to have an effect. Getting something that large spinning takes significant amounts of energy and time. Getting something that large to stop spinning can be equally problematic. Yes, you could use brakes (and many do), but you need to dissipate a lot of energy (heat).

So, taking into account that these gyros don't change speed very fast, and that they very heavily counter forces, you have a number of problems off the bat:


You have to carry a LOT of extra weight.
You need to have a lot of extra fuel/motors/generators to get the mass spinning.
Once spinning, the mass will counter changes in momentum, this will make the aircraft difficult, if not impossible, to bank (aka: turn).


So could they be used to correct a flat spin? Not really. In order to counter the flat spin you'd have to have a gyro mounted 90° to the angle of the spin, and then change the momentum of the gyro to counter the spin. Flat spins are not (apart from what Top Gun may have you believe) impossible to recover from given altitude and control authority. This happens in the world of aerobatics quite frequently (and on purpose).

Can they be used to control the plane? Again probably not. You are talking about changing the angular momentum of a huge mass very quickly. This would work for the braking side, but not the speed-up side. That means you'd have an aircraft that can perform a maneuver once in one direction, then have to fly off while the aircraft builds up gyro's again. 

By the way, spacecraft use control moment gyros that work by using gimbals to rotate a gyroscopic flywheel to impart a rotational force on the spacecraft. These work because of a reduced gravity/drag in the operational environment of the spacecraft itself. The masses don't have to be as big as they would on earth.