There are no general rules for buoyancy in either rule set
(Quoted passages below are from the 2024 rules unless otherwise noted, but the 2014 rules are similar enough on the relevant points that all the same logic holds.)
You are probably thinking of the rules for flying, which do explicitly have a consequence for having your speed reduced to 0:
While flying, you fall if you have the Incapacitated or Prone condition or your Fly Speed is reduced to 0.
Given this, it would be reasonable to imagine a similar mechanic for rule swimming, which would cause you to sink if your speed is reduced to 0. However, as far as I know there is no such rule. The description for swimming is as follows:
While you’re swimming, each foot of movement costs 1 extra foot (2 extra feet in Difficult Terrain). You ignore this extra cost if you have a Swim Speed and use it to swim. At the DM’s option, moving any distance in rough water might require a successful DC 15 Strength (Athletics) check.
There are also additional rules for underwater combat, but these do not mention sinking or floating. Notably, the description of the Athletics skill does mention floating in the "example uses" column:
Jump farther than normal, stay afloat in rough water, or break something.
However, I can't find anything in the rules that gives a general rules for when a creature sinks or floats, although a few spells and effects do specifically influence a creature's ability to float, such as the 2014 version of Water Walk, which specifies:
If you target a creature submerged in a liquid, the spell carries the target to the surface of the liquid at a rate of 60 feet per round.
(Incidentally, the 2024 version of the spell has changed a bit and no longer has this effect.)
As such, outside of these few specific cases, determining when a creature sinks or floats is left almost entirely to the DM. And notably, the reference to using the Athletics skill to stay afloat shows that the rules do indeed expect the DM to determine what does or doesn't float in the course of normal gameplay involving bodies of water.