No, and for a delightful discussion of this popular myth, visit the Bad Astronomy web site for a great essay on this. This effect can only be weakly measured under very careful, controlled laboratory conditions with carefully designed basins of water. You have to let the water sit for weeks in the basin because the slightest turbulence and biasing from how it was filled, completely overcomes the weak vorticity. Nothing as clumsy as water down a sink or toilet will work.
All answers are provided by Dr. Sten Odenwald (Raytheon STX)
NASA IMAGE/POETRY Education and Public Outreach program.