I don't see "do" as being spiritual (in the sense of spirit being something ethereal or mystical, some outside force or concept that one internalizes - my definition), but rather something a bit more concrete. Funakoshi wrote Karate-do, My Way of Life and that title I think best describes "do" - the way you walk the path of everyday life using the concepts and methodologies of karate. "Do" is dedication, commitment, respect, self-discipline and improvement and morality.
Funakoshi's book came out in 1923 (don't know when "do" was first used, but this made it popular), the time karate was being established into the Japanese school system. Karate started to change from pure self-defense (jutsu) to a system of self-development and behavioral model as befitting an educational venue molding boys and girls. This grew into the modern popular view of karate.
While karate-jutsu developed many of the virtues of "do," they were a natural by-product of the art, not a major focus of it as in karate-do.
One can insert/visualize spirituality into their karate practice, like anything else, if they wish, but it's not an inherent part of it.