Ok, so I'm a legal nerd. Let me just get that admission out of the way in the beginning. In that vein, I have actually studied the decision cited by Buck, Alexander v. Mineta, 531 U.S. 940 (2000). Actually, that case is a one-sentence affirmation of a decision by the D.C. District Court, which is reported at 90 F.Supp.2d 35 as Adams v. Clinton (two cases were combined because they concerned the same issue).
It's always important when looking at legal precedent to understand the facts of the case. These two cases, for example, were suits by citizens of D.C. and the District itself which claimed that it was unconstitutional to deny them the right to vote for representatives in Congress. While the opinion is a fascinating treatise on that subject, it actually says very little that is on point to our current debate.
The relevant part of the case is the discussion on whether it is constitutional for a statute of Congress to limit a pre-existing right of D.C. citizens to vote for reps in Congress. The case held that the relevant statute, the Organic Act of 1801 that created D.C., did not do that, and that in fact the Constitution itself was the source of the D.C. citizens' disability to vote for Congress (the Constitution provides that citizens of the states will so vote, and D.C. is not a "state"). The opinion agreed that a statute could not have validly taken the vote away from them ("Thus, it was not the Organic Act or any cessation-related legislation that excluded District residents from the franchise, something we agree could not have been done by legislation alone").
Implicitly recognized, even by this decision, then, is a constitutionally defined right to vote. Whether and how a state can reasonably regulate that right has been the subject of numerous other decisions. The following represents a few:
"Once the franchise is granted to the electorate, lines may not be drawn which are inconsistent with the Equal Protection Clause of the Fourteenth Amendment. That is to say, the right of suffrage is subject to the imposition of state standards which are not discriminatory and which do not contravene any restriction that Congress, acting pursuant to its constitutional powers, has imposed." (for example, the Voting Rights Act -ed.)
"We conclude that a State violates the Equal Protection Clause of the Fourteenth Amendment whenever it makes the affluence of the voter or payment of any fee an electoral standard. Voter qualifications have no relation to wealth nor to paying or not paying this or any other tax. Our cases demonstrate that the Equal Protection Clause of the Fourteenth Amendment restrains the States from fixing voter qualifications which invidiously discriminate."
Harper v. Virginia State Board of Elections, 383 U.S. 663 (1996) (ruling unconstitutional a $1.50 poll tax).
"A government unit may legitimately restrict the right to participate in its political processes to those who reside within its borders." Holt Civic Club v. Tuscaloosa, 439 U.S. 60 (1978).
"Cases invalidating governmentally imposed wealth restrictions on the right to vote or file as a candidate for public office rest on the conclusion that wealth "is not germane to one's ability to participate intelligently in the electoral process" and is therefore an insufficient basis on which to restrict a citizen's fundamental right to vote." Buckley v. Valeo, 424 U.S. 1 (1976).
"[States have] unquestioned power to impose reasonable residence restrictions on the availability of the ballot. There can be no doubt either of the historic function of the States to establish, on a nondiscriminatory basis, and in accordance with the Constitution, other qualifications for the exercise of the franchise. Indeed, "the States have long been held to have broad powers to determine the conditions under which the right of suffrage may be exercised." Lassiter v. Northampton Election Bd., 360 U.S. 45, 50. Compare United States v. Classic, 313 U.S. 299; [***678] Ex parte Yarbrough, 110 U.S. 651. "In other words, the privilege to vote in a State is within the jurisdiction of the State itself, to be exercised as the State may direct, and upon such terms as to it may seem proper, provided, of course, no discrimination is made between individuals in violation of the Federal Constitution."
"We deal here with matters close to the core of our constitutional system. "The right . . . to choose," United States v. Classic, 313 U.S. 299, 314, that this Court has been so zealous to protect, means, at the least, that States may not casually deprive a class of individuals of the vote because of some remote administrative benefit to the State."
Carrington v. Rash, 380 U.S. 89 (1965)(striking down Texas law denying soldiers the right to vote in Texas).
All emphasis in the quotes added by me. Bottom line, yes, of course a state can restrict the right to vote to those who are qualified, but its means of doing so must be related to the qualifications and must not discriminate in violation of the Constitution. Putting aside and assuming for the moment that an ID requirement is rationally related to, as you put it, proving the person is who they say they are (and therefore a resident and citizen qualified to vote), the problem with the Georgia bill is whether it discriminates in violation of the Constitution against those who can not afford to purchase the ID (I realize you claim they are free, but I will incorporate by reference Jonathan's argument on that point).
I would have to do a whole separate post to talk about the Voting Rights Act, which I don't have time for and haven't done the research for, but suffice it to say for now that the test is whether there are "discriminatory effects" on the basis of race (and maybe class too? like I said, haven't done the research yet on that one). If there are (which isn't like, you know, outside the realm of possibility, considering the still-lingering correlation between income, wealth, and race, particularly in the South), then the bill may very well violate the VRA, even if it does not constitute an equal protection violation as set out above.
Sorry, folks, but the lawyer in me couldn't resist it this time.