Assuming that mercury is more dense then water (I am taking your word on that for my answer though from my memory that is correct) then you could definitely swim on it, you may not even need to swim as you stated. Some people (no offense but fat people, me included) can float on top of water already as our density is less than that of a smaller or more muscular person. The exact calculations to determine what density of person could float on what density of liquid is actually quite extensive and far beyond my current knowledge but yes, you could swim in mercury.
Now, to answer the question originally asked though I am sure it isn't what you really wanted. Can you drown in mercury, yes, of course. Really, you could drown in any liquid. Drowning is simply what happens when your lungs fill with something in a liquid form thus depriving your body of oxygen.