Actually those sections of the Constitution don't simply say votes can't be denied based on certain charactersisticss, they say the right to vote can't be denied.
Amendment 15
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.
Amendment 19
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.
Amendment 24
The right of citizens of the United States to vote in any primary or other election for President or Vice President, for electors for President or Vice President, or for Senator or Representative in Congress, shall not be denied or abridged by the United States or any State by reason of failure to pay any poll tax or other tax.
Amendment 26
The right of citizens of the United States, who are eighteen years of age or older, to vote shall not be denied or abridged by the United States or by any State on account of age.
So ya, "the right of citizens of the United States to vote" is in the Constitution.
>>>>
Well kinda, sorta...but not really. Those are applicable only when the state has granted the right to vote to begin with. In reality all of them have so it's more of an academic question rather than a practical one. But lets have a closer look:
The 17th Amendment uses the word "elected" and therefore indicates voting rights in regards to the Senate:
"
The Senate of the United States shall be composed of two Senators from each State, elected by the people thereof, for six years; and each Senator shall have one vote."
Article I, Section 2
suggests voting rights in regards to the House of Representatives:
"
The House of Representatives shall be composed of Members chosen every second Year by the People of the several States"
The language "chosen by the people" is ambiguous. All the people? Some of the people? How do the people choose? What's the process? The Constitution does not spell that out but leaves it in the hands of the individual states to determine. Indeed in the beginning, only land owners and the wealthy were afforded the right to vote. As time passed and political parties came into being each side needed more supporters and so voting rights were reluctantly extended to other groups of people, BUT
those were state decisions not federal. Just as one side needed certain groups to have voting rights, the other side often wanted that group to remain "voteless". That's when the Federal government got into it and said "no you can't deny voting based upon age, gender, race, etc.
If you are going to use voting as your process of choosing, you have to let those people have a vote as well".
But the key part of that is: "
If you are going to use voting as your process of choosing"
Let's look at Article II, Section I which states in regard to choosing the President:
"
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: but no Senator or Representative, or Person holding an Office of Trust or Profit under the United States, shall be appointed an Elector."
The 12th Amendment alters some language and says:
"
The Electors shall meet in their respective states and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same state with themselves;..."
So what this essentially says is that the states can decide for themselves how their electoral votes are cast. This is why Maine and Nebraska can allocate their EVs somewhat proportionally while the rest of the nation is winner take all. They have the right to make that decision for themselves.
Furthermore, any state could get a wild hare up their ass and say "you know....**** this voting bullshit. We are just going to have the governor decide which candidate gets our EVs" and there would be no violation of the United States Constitution. Now without question there would be outrage by the people of the state and the state officials would be cutting their own political throats. The state Legislature in most cases (maybe all of them) would have to amend the state constitution, and Congress would likely immediately amend the US Constitution to stop that from happening, but as it stands now....any state could do that if they wished.
What this means is that people have the right to vote for President
only if the state gives it to them, they have the right to vote for Senators, and they have the right to participate in the choosing of Representatives in whatever way the state determines.....could be a statewide game of lawn darts if they so decided and indeed some states have similar things in their voting laws. In New Mexico, for example, ties in an election are broken by a game of poker between the tied candidates.
Now again, this is purely academic since all states grant the "right to vote" to their citizens and use voting as the method for choosing their Senators, Representatives, and directing their Presidential Electors. But they don't
have to do it that way and indeed the Electors do not
have to cast their EV according to the results of the election. In fact, there are several examples throughout United States history where electors have ignored the results of the vote and cast their ballot the other way.
http://www.pokerpages.com/poker-news/news/game-of-poker-settles-tie-in-new-mexico-election-30535.htm
http://www.archives.gov/exhibits/charters/constitution_transcript.html