As far as I'm aware this has never been attempted (and I doubt the Navy would let me try - though I'm certainly game if they are!), but: A Piper J3 Cub stalls at 38mph (33kts).
The USS Enterprise (CVN-65) had a maximum speed of 38.7mph (33.6kts).
So, given a lightly loaded cub and an Enterprise -class carrier at flank speed, it is at least theoretically possible to float that aircraft off the deck (if it were lightly loaded, and thus had a lower stall speed).
Achieving zero relative motion would require a little more effort - specifically you'd have to seek out a headwind, and the Cub would have to be operating at an airspeed above stall, but still pretty close to it.
For example, if the Enterprise were making 30 knots and steaming into a 10 knot headwind the relative wind on deck would be 40 knots.
That's enough to lift the Cub, but as soon as it was in the air it would need to pull itself along (via its engine) to maintain that speed.
The Cub would have zero relative motion above the deck at an airspeed somewhere around 40 knots).
As a (slightly ridiculous) bonus:
The Belite Superlite has a stall speed of 28 mph (25kts).
It would be theoretically possible to fly such a plane like a kite from an aircraft carrier.
Its advertised takeoff and landing distances (both ~500ft) are also such that it could operate from a carrier deck without need for the catapult to launch or arresting wire to stop.
aviation.stackexchange.com/users/64/voretaq7
A Cessna 150 can take off at 48 knots rotation speed.
An aircraft carrier can do 30 knots, and an 18 knot headwind is certainly possible at sea.
That puts a 48 knot wind over the wings, enough for the Cessna to take off.
The engine would counteract drag, allowing the plane to "float" off the deck.
If you have any doubt about this being possible, just watch what happens if you don't tie down your plane properly.
http://www.youtube.com/watch?v=IPOtDPHjW-Y
aviation.stackexchange.com/users/1696/fooot
What you say is not only possible, but it has been used to land a plane on a much smaller (and a bit faster) ship than an aircraft carrier.
Try to search in the web for the video
"Extreme small plane landing on a ship at sea".
Here is a working link at the moment: http://www.youtube.com/watch?v=pUdzVnZBaoY
aviation.stackexchange.com/users/2345/DarioP
It all depends on the minimum flight speed and the ship's maximum speed.
Even with the slow speed of submarines, floating a flying machine is possible if the machine has been designed that way.
In order to increase the observation radius of submarines (typically only 5nm), an unpowered gyrocopter was designed in 1941 to lift an observer 500 ft up into the air to increase the observation radius to 25 nm.
Gyrocopters are known for their low flying speeds, and this one, the Focke-Achgelis 330 , was designed to be folded away in a watertight container and to be stored on the deck of the submarine.
It could be assembled and de-assembled within 20 minutes and was then towed behind the submarine like a kite.
The minimum flight speed was only 27 km/h (14.5 its), just enough for a submarine going at full speed and a little wind.
aviation.stackexchange.com/users/1961/Peter Kämpf
The forward speed of a carrier might not be the listed max speed, if heading into a strong headwind.
However, if the relative wind is great enough, the aircraft wings would create more lift than the weight of the plane.
There are of course issues like structure induced turbulence which might adversely impact handling.
And if near equilibrium, the aircraft may never fly beyond the bow.
If planning on this type of operation, would look for a Vx (adjected for engine off or idle) relative wind to allow a reasonable climb (if you call the ascent that) to clear the structure.
Also, I would plan on factoring in the ground effect of working off the carrier deck, and then having that effect reduced as altitude was gained.
Finally, from practical experience, I would expect the aircraft to become airborne prior to the stall speed, due to ground effect, probably a lighter that gross weight loading, and lots of other factors.
aviation.stackexchange.com/users/18572/mongo
Would it be possible to float a small plane (say a  Cessna 150 ) off of the deck of an aircraft carrier without the Cessna achieving any forward motion with respect to the aircraft carrier? If not a Cessna 150, is there a standard GA aircraft that could do it?  EDIT:  Per a comment from ratchet freak: the takeoff would be done via lift from the wings, not via an engine duct (VTOL), like a  Harrier .
Can a small plane be floated off of an aircraft carrier?
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<takeoff><general-aviation><aircraft-carrier>
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aviation.stackexchange.com/users/511/Jay Carr
It is possible to float a small plane off an aircraft carrier. It has been done to land a plane on a smaller faster ship. Success depends on the minimum flight speed and the ship's maximum speed. Not only is this action possible, it has already been used to land a plane on a ship that is much smaller and a little faster than an aircraft carrier. Whether or not it is possible in the individual circumstances depends on the minimum flight speed of the plane and the maximum speed of the ship. As a specific example, an unpowered gyrocopter was designed in 1941 in order to lift an observer into the air to increase the observation radius of submarines.
Not only is this action possible, it has already been used to land a plane on a ship that is much smaller and a little faster than an aircraft carrier. Whether or not it is possible depends on the minimum flight speed of the plane and the maximum speed of the ship. In 1941, an unpowered gyrocopter was designed in order to lift an observer into the air to increase the observation radius of submarines.