Actually, there is strong precedent in Canon.
Voyager had no issues with Fluidic Space.
And the Enterprise in The Immunity Syndrome was essentially swimming inside a giant amoeba, certainly a liquid environment roughly analogous to water.
The functioning within a 1G environment is slightly more problematic, until you realize that a starship is travelling at warp velocities, and has survived close proximity to far more intense gravitationally induced stresses, to the point of surviving time travel.
Can't really use Fluidic Space as an example, because we really have no idea of the pressure and viscosity of that particular fluid.
But it probably still wouldn't have any trouble, at least not if it moved slowly. Any ship that can survive the streses of a) skirting the event horizon of a black hole and b) getting blasted out of said black hole by a massive antimatter explosion probably isn't going to have any issues with water pressure... so long as it doesn't dive very far.
Does anyone know if NASA uses (or rather, used) submersion tests in order to determine the structural integrity of things like shuttles and Apollo capsules?
They use space suits when submerged in water for training for Zero-G missions.
If present-day spacesuits can easily deal with a fluid environment, then the extemes and rigours of warp travel, interstellar environments, Ion storms, extreme gravity (such as a "black star" or black hole), Quasar-like objects (The Galileo Seven) and giant amoebas among other strange, unpredictable situations, taking a trip in a Class-M planetary ocean body should not be too much of a stretch.
It should also be noted that the ship is known to survive extremely low orbital flight, as Captain Christopher and some US Air Force personnel can attest.

This kind of situation would likely prompt scotty to worry about the effects on his "Poor Bairns".