Surface Tension Support for FDM Printers
It's not feasible as described with normal FDM technology.
FDM bases on depositing material in a single path. This needs the deposited material to stay at the same XY coordinates for subsequent paths. And exactly here is where a floating piece fails: a free-floating piece is by its very definition unrestricted in XY, and would move to follow the nozzle.
There are is also a whole plethora of factors that make this idea not feasible with the standard technology, meaning you'll have to develop the whole process, not just recombine two ideas. This means, you need to solve the following issues:
Heatsink Water
The extruded plastic needs to stay close to the melting point for some time, so it can fuse and bond with the lines next to them. However, water is known to be a very good method to get the heat away from items, as each liter of water can take about 4.1 MJ and only heat by one Kelvin. PLA on the other hand only stores about 1.8 MJ per Kilogram and Kelvin.
As a back of the envelope calculation, the temperature differential between room temperature and printing temperature PLA is about 180 Kelvin. Each gram of PLA is equivalent to 319.8 mm of filament (assuming a density of 1.3 g/cm³) or an extruded line of 9.6 kilometer length of 0.4 mm width and 0.2 mm height! That one gram contains about 324 Joules of energy that will be dispersed to the room temperature as it cools down. The water vat would not even get measurably warmer from sucking those few joules from a whole print!
While this could be, in the right setup, be used to rapidly cool the print and solidify it in shape, the result of the rapid healing will most likely also impact print quality negatively, as cross-layer bonding is reduced.
Separator water
It is a well-known trick in creating polymer fibers to extrude underwater, as the water not only cools (see above) but also acts as a separator between the fibers, for the very short timeframe they are still malleable. This would also strike when printing into the water - there'll be a water layer in between the deposited filament, which would need to either get pressed out or cooked out before any cross-extrusion bonding can occur. As a result, just extruding into the water should result in a print that has almost no sturdiness, and might fall apart on touch.
Floating
Water has a density of 1 g/cm³. PLA has a density of 1.3 g/cm³. So a solid chunk of PLA sinks. But we don't print solid, we include air. Not just a few percent but infill is usually below 20 %. I have just printed a cube. After smoothing the surface by sanding, the cube is 29,7x29.9x29.9 mm. It has 3 parameters, 20% infill, 5 top and bottom layers with 0.2 mm layer height and comes to 11 grams. 11 g/26.55 cm³=0.41 g/cm³. Or in other words: the cube would float, about 40 % under the water surface, 60 % above the surface. The print would be, as a result, quickly break the water surface and get no support from the water at all.
Submerging the bed?
The main issue of a free-floating object (position) might be mitigated if there was a bed that would be submerged, but one would open a new can of worms, that might be even worse: the volume of the print and the accuracy of pumps.
As the print goes on - quality be dammed - the print grows in volume. However, it doesn't grow entirely linear, depending on two factors:

Is the print happening on/below the water surface? Then we will include water in the print. The volumetric growth of the print is in this case just the deposited filament. You'll need an overflow to compensate for the print growing and keep the water level in position.
Is the print happening just above the surface of the water? then the displaced volume grows according to the depth of the print in the water. This would mitigate most of the problems from water preventing cross-bonding and causing floating, and even use the heatsink properties more beneficial, as any filament starting to sag will be stopped. It also would prevent water from being encased in the print. However, it does not use surface tension. Also, you'll need an overflow system to keep the water level steady.