The comments about the all volunteer force being important for the military culture are on the mark.
Additionally, it would be highly impractical for the United States to set up a compulsory military service structure similar to the old German Wehrpflict. What you have to understand about these systems is that they are tremendously hard to maintain. They require adjudication officers and tremendous amounts of administration and supporting infrastructure.
These things may be doable in a relatively small and crowded country like Germany. However, the United States has three and a half times the population of Germany living in over 26 times as much land area. Who is going to chase truants into the hills of Idaho or Tennessee? Who wants to pay for all of that transportation?
Besides all of that, is it really a good idea to force all Americans into the same mold? Wouldn't that engender an atmosphere of group-think? I think our diversity of thought, backgrounds, and skills is a real blessing for this country.
It is easy to make a knee jerk statement, and say that compulsory military (or public) service would be good for America. I believe that such a system would have deleterious effects on the country.