In nations that do in fact have a national ID, everyone gets one, with little or no effort (certainly no need to travel), and if you wanted to mark the card "non-voting" that wouldn't be a big step.
Your right to vote is established by the US Constitution. A state can't burden that right. If a state wants to add an ID requirement, that's fine, but getting the ID must present no burden to any voter.
In reality, the conundrum here is entirely of conservatives' own making. On the one hand, they (non-exclusively) are adamantly against automatically issued national or even state IDs. On the other, they want to require ID for voting (ascribe whatever reason to this you want). You can't fulfill both these desires without violating the constitutional right to vote.