Of course you can be a witch with no religion. Nature is the true source of all magick. Just look at it! Every deity is an aspect of nature...apart of nature! Therefore nature is the "whole", and "deities" are the "spirits of", the parts!
Every religion breaks it down differently, but all in all, it's still the same basis.
Look at the tree of life. It is made up of divine (branches), mundane (ground level) and spirit (roots). But the tree itself--life itself--encompasses all three.
Never do anything that doesn't feel right to you. Keep an open mind and follow your heart, and you'll walk down your personal path. There is no need for labels. There is no need to "fit into" other people's beliefs. Belief is so very personal, my viewpoint on it is that it should be discovered and deciphered by the individual only, with the Higher Self and common sense as the only guidance.
The Oracle at Delphi's "Know Thyself" is a powerful statement. How can we know ourselves by looking around us for answers? We can't. We will only know ourselves truly by looking within.