The whole premise of the constitution is that it defines some basic civil rights. Your hypothetical is invalid, because the right to vote is a civil right.
Really? I don't remember the words "civil rights" anywhere in the Constitution. I thought it was a framework for how the government was intended to be architected and definitions of what the government is supposed to be responsible for.
Even Amendments 15, 19, and 26, which speak directly to the right of voters and guarantees the right to vote
to citizens does not prevent non-citizens from voting:
The right of citizens of the United States to vote shall not be denied or abridged by the United States or by any state on account of race, color, or previous condition of servitude.
The right of citizens of the United States to vote shall not be denied or abridged by the United States or by any state on account of sex.
The right of citizens of the United States, who are 18 years of age or older, to vote, shall not be denied or abridged by the United States or any state on account of age.
The language does not preclude non-citizens from voting; they simply say that these characteristics are not to preclude citizens from voting. For instance, California, Alaska, and Florida all have had movements to lower the voting age to 16; while none were successful, it was not because it would have been illegal. States are free, as sovereigns, to determine who are qualified electors.
To clarify my meaning of the above: taken in whole, Amendments 15, 19, and 26 should be read to mean "The right of citizens of the United States, who are 18 years of age or older, shall not be denied or abridged by the United States or any state on account of age, sex, race, color, or previous condition of servitude." It does not mean, "Only citizens of the United States, who are 18 years of age or older, shall be qualified electors regardless of age, sex, race, color, or previous condition of servitude."
Another interesting case is the Presidency. Most Americans think that it is their right to vote to elect the President; they are in fact mistaken. It is the States' rights to elect the President, chosen by the Electors appointed by the States' choosings. While most states dedicate their entire electoral vote to the plurality winner of the state (because otherwise it would be extraordinarily difficult to have a 50% + 1 vote), some do not (I believe Indiana is one such exception to this rule). If you don't like the way this is set up you are free to petition your state to change it. Proof that it's a state right?
Each state shall appoint, in such manner as the Legislature thereof may direct, a number of electors, equal to the whole number of Senators and Representatives to which the State may be entitled in the Congress
