I would probably say something like:
- Since water is neutral on its own, dissolution of an ionic solid (like the salts in solutions X, Y, and Z) will only alter the pH from neutral (pH = 7 at 25 oC) if one or more of the resulting dissociated ions hydrolyses (reacts with water) to produce additional hydronium or hydroxide ions.
- The ammonium nitrate solution will be acidic due to hydrolysis of its cation: NH4+(aq) + H2O(l) <----> NH3(aq) + H3O+(aq). The resulting hydronium ions will cause the pH to fall below neutral and so solution Z is ammonium nitrate.
- The sodium acetate solution will be basic due to hydrolysis of its anion: CH3CO2-(aq) + H2O(l) <----> CH3COOH(aq) + OH-(aq). The resulting hydroxide ions will cause the pH to increase above neutral and so solution Y is sodium acetate.
- The sodium chloride solution will be neutral as neither of its ions will hydrolyse and thus there is no chemical process that leads the pH to change from neutral. Thus, solution X is sodium chloride.
The reason why nitrate cannot (or, more accurately, does not) hydrolyse is that the product formed would be nitric acid, a strong acid, which would immediately ionise into a nitrate ion and a hydronium ion, thereby neutralising the hydroxide ion. Put another way, since nitric acid is a strong acid in water, its ionisation
HNO3(aq) + H2O(l) ----> H3O+(aq) + NO3-(aq)
is not an equilibrium nor is it reversible. So, if the nitrate anion has such poor basic properties that it cannot be protonated by hydronium, then there is no reason to expect that it could be protonated by water.
In each of the systems that appear in my answer, the resulting molecular product is a weak acid or base and so its presence in an aqueous solution is reasonable and not unexpected. The presence of unionised nitric acid molecules in an aqueous solution would be unexpected. Its presence in proximity to hydroxide ions, such as is suggested by an equation like
NO3-(aq) + H2O(l) -X-X-X-> HNO3(aq) + OH-(aq)
would be remarkable if it occurred to any significant extent.
Am I saying that it is impossible for a collision between a water molecule and a nitrate ion to result in a proton transfer reaction and the formation of an unionised nitric acid molecule and a hydroxide ion?
NO. It is not impossible for this to occur... but it is improbable.
Am I saying that it is impossible for such a process to result in a nitrate-containing salt to have a basic solution (ie. significantly above the neutral pH at the relevant temperature)?
YES. If a nitrate-containing solution is basic, look for an explanation that does not involve nitrate hydrolysis. The more basic such a solution was, the less likely for unionised nitric acid molecules to persist in solution for longer than moments as the comparatively high concentration of hydroxide ions would rule out any possibility of unionised nitric acid molecules remaining unreacted.