Concept:
The acidity of a substance depends on the solvent's ability to accept protons (basicity). Liquid ammonia is a more basic solvent than water, which influences the ionization of acids dissolved in it.
Step 1: {Compare the basicity of solvents.}
Ammonia (NH$_3$) is a stronger base than water (H$_2$O). This is because the lone pair on the nitrogen atom in ammonia is more available for protonation than the lone pairs on the oxygen atom in water.
Step 2: {Analyze the leveling effect.}
In a highly basic solvent like liquid ammonia, even relatively weak acids like acetic acid are forced to donate their protons completely.
$$\text{CH}_3\text{COOH} + \text{NH}_3 \rightarrow \text{CH}_3\text{COO}^- + \text{NH}_4^+$$
Step 3: {Determine the relative acid strength.}
Because liquid ammonia accepts protons more readily than water, acetic acid ionizes to a much greater extent in ammonia. Therefore, it behaves as a stronger acid in liquid ammonia than it does in water.