Is there any publicly accessible record that would allow one to confirm whether a person is currently or has served in the U.S. armed forces?
No.
Unless the person was especially good (and got medals, etc.) or especially bad (and got court-martialed).
The official DD-214 form is only available to the servicemember, or their next-of-kin.
Many units (especially Navy) have websites now, with listings of all the people who served on that boat. But nothing official about these, and accuracy varies.