Unfortunately it does. Hasn't been called up since the Vietnam War but the system is still in place and people are still required to register.

https://www.sss.gov/

No, it doesn't.

> Hasn't been called up since the Vietnam War

Right, because there is no draft law.

> but the system is still in place and people are still required to register.

No, the registration system is in place, but it would take a change in law for there to be a draft.