The Red Eye Tool desaturates the center portion of the selection that you create by clicking-and-dragging the tool around the eye. Sometimes you may need to create a second selection to remove the red color from the eye.
A better approach would be to create a new layer, set the blending mode to color, and paint with black using the brush tool. This will make the changes non-destructive, and give you far more control over the area you desaturate. If the eyes are too light, duplicate the layer and set the blend mode to multiply, and bring down the opacity accordingly.