It's a tricky question, some fungi would survive without any trouble, and some parts of all fungi would survive. The reason you don't "see" a given organism (single celled) in water is often that it can't compete well under those conditions, not that it can't survive there.
Fungi evolved from aquatic species (as we did), and there are many groups of fungi that thrive in water, like the members of Oomycetes (Saprolegniales, Leptomitales and Lagendiales) and Labyrinthulomycetes. Some fungi would die when exposed to air, they are adapted to oxygen-free environments.
The fungal groups you normally associate with fungi (Basidiomycetes, Ascomycetes, Discomycetes etc.) would not have effecient dispersal under water, since their reproductive organs (like the mushroom "hats" from basidiomycetes) are adapted for dispersal via air.
In water the diffusion rate of oxygen is much much slower than in air, this is the reason that some organisms would die when submerged in water, but the diffusion rate is usually sufficient for single-celled organisms or organisms that are not more than a single cell "thick" anywhere, like the mycelium of multicellular fungi.
The answer to your question should be that if a fungus is adapted to function in an aérob (oxygen rich) environment, then it would probably be fine if it was single/few-celled or membran like, but the larger it is the less oxygen would be able to diffuse to it's 'core' and the threat of dying due to oxygen depletion would increase with the size (volume).
It does however make a huge difference whether it's salt water or not. Many organisms would be sucked 'dry' via osmosis in salt water, including human cells (we evolved from fresh-water fish).
