Window, New Image for ...
Go to Image > Duplicate and close the original.
Zoom one of the windows so that you can see the eyes
as large as possible. Set the other window view to 100%.
Arrange the two windows so you can see both the
zoomed view and the 100% view at the same time.
Create a new layer. (layer, new layer)
Use the eyedropper to pick up a color from the iris
of the eye. It should be a fairly gray tint with a hint of the eye
color.
Go to the Paint Brush Tool. Change the size to 1 or 3
Paint over the red part of the eye on the new layer,
being careful not to paint over the eyelids.
Go to Filters > Blur > Gaussian and give it about a 1
pixel blur to soften the edges.
Set the layer blend mode to Saturation. This will
take the red out without removing the highlights, but in many cases it
leaves the eyes too gray and hollow looking.
If that's the case, duplicate the saturation layer
and change the blend mode to Hue. That should put some color back in
while still preserving the highlights.
If the color is too strong after adding a Hue layer,
lower the opacity of the Hue layer.
When you're happy with the results you can merge the
extra layers down.