Why Do Most Humans & Cats Have Blue/Bluish Eyes At Birth?
The blue colour is an example of structural colour, caused by light interacting physically with something. Some examples of structural colour are the iridescence of insect wings and body surfaces (usually caused by repeating chitinous structures), and of certain birds feathers, as well as the non-iridescent colours of the blue and yellow macaw.

In the case of the eye it is the stroma of the iris that is responsible. This is a network of fibrous tissue which scatters light. When light is scattered in this way (Rayleigh scattering) it is the short wavelengths which are most scattered. So when you look at the sky (away from the sun) you are seeing diffuse blue light created by this scattering in the atmosphere. In the case of the iris what you are seeing is light reflected from the eye being scattered within the iris, creating the diffuse blue colour. As you say in the question, everything else is down to melanin.