Evaporating the water of a solution of only carbonate ions and water gives?
I would love to add some quantum chemical bone to the stew we are trying to cook here.

Let us assume that it is possible to have a solution of carbonate ions, like already discussed by Nicolau in the comments. Then Rauru is also right that you will have to consider several equilibria:
\begin{aligned}\ce{
(1) &amp;&amp; 2H2O &amp;&lt;=&gt; H3+O + {}^{-}OH \\
(2) &amp;&amp; CO3^{2-} + H2O &amp;&lt;=&gt; HCO3- + {}^{-}OH\\
(3) &amp;&amp; HCO3- + H2O &amp;&lt;=&gt; H2CO3 + {}^{-}OH 
}\end{aligned}

Now $(3)$ seems a little far fetched, so we will ignore it.

When you do now remove water from the equilibrium through (gentle) evaporation, you will have a mixture of only $\ce{CO3^{2-} + HCO3- + {}^{-}OH}$. Further assuming this is possible you will always have some collisions within this ensemble. Statistical it is therefore always possible to also remove $\ce{HCO3- + {}^{-}OH}$ via $(2)$ from the system. Also single entities of carbonate ions will repel each other as states by G M. So basically it all boils down to on question: Is carbonate stable in vacuum? What happens to carbonate if you put it into vacuum?

Now we have some quite good approximations to these particular conditions (gas phase and 0K), but I do not want to go into detail. I did a calculation on the carbonate with Gaussian09 and as a cheap shot BP86/cc-pVTZ level of theory. The only thing I want to show is the highest occupied molecular orbital.



What can clearly see is that this is no bonding orbital. There are lone pair contributions from all Oxygen atoms (due to imposed symmetry constraints). In addition to the high negative charge, electrons will not be stabilized by the field of the nuclei. Also, Orbitals only vanish at infinity, giving electrons a non-zero probability to actually leave the compound. Furthermore, there are two electrons in this orbital, so the spins of these are paired, which also costs energy.

My suspicion therefore is that the carbonate at least loses one electron. To where? To anywhere. (Also mentioned by Nicolau in the comments.) I do not have any actual proof for this, as it quite hard to find people actually doing experiments and/ or calculations on that topic. I remember that one question of a Professor of mine in the first semester was: "What happens to sulphate if you shoot it into space?" The answer was given without any further explanation and I never bothered to ask: "It looses an electron."



Note:

Another point that should always be considered when dealing with gas phase reactions is, that when ions have enough kinetic energy they might react in the strangest ways. One of them might be
$$\ce{2CO3^{2-} -&gt; 2CO2- + O2^{2-}}.$$

Also collision with any particle may cause electrons to be transferred.