Thanks to some helpfull comments I could find the problem with the texture. I still had it on Generated instead of on UV. Also the texture was facing the wrong way so I made a new one.
The eyeball has both a generated texture and a UV texture.
The UV texture is for the veins.
The generated texture makes the cornea transparant and makes it blend into the white part of the eye in the same way as it looks on a real eye. Most tutorials I've seen don't do this causing a hard edge between the cornea and the white part of the eye which looks rather unrealistic.