No, the white color of the paper never counts as a color, it is simply because of the absence of the ink. However, if you use color paper, then the white ink can be applied if any text or graphic is required.
However, it also depends on whether we are applying inks on white paper or any other colored material. If the paper is white then there is no need for white color. White is considered a color if you are printing with white ink on any material.
