No, gravity holds it down just like it does us.
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No it does not
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The rotation in the major ocean basins is driven by a combination of wind stress at the ocean surface and the Coriolis force due to the earth's rotation. As discussed in the unit on atmospheric structure and circulation , the winds at the earth's surface are directed from east to west in at the Equator and generally west to east at the middle latitudes (30 to 60o north and south of the equator). The wind induces an ocean drift current generally in the same direction but rotated slightly, deflected by the Coriolis force to the right of the wind direction in the Northern Hemisphere and to the left of the wind in the Southern Hemisphere.

The law of conservation of momentum says that in the absence of forces, the momentum of an object does not change. A parcel of water (or air) that has a component of its velocity toward the Equator (outer boundary of the circle) will be moving to positions at greater distance from the polar axis. Because of the rotation of the earth, the total speed of the parcel will increase (and violate the law of conservation of momentum) unless it moves in the direction opposite to the rotation, as shown, and follows a path that curves to the right. A parcel moving towards the axis of rotation (northward in the Northern Hemisphere) similarly will move in a clockwise direction. For flow south of the Equator, the same reasoning leads to the conclusion that rotation in the Southern Hemisphere is in the counterclockwise direction.

Read more :- http://encarta.msn.com/sidebar_761599222/giant_ocean_cataracts.html
Source(s):
http://www.geology.iastate.edu/gccourse/ocean/ocean_lecture_new.html

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