Carbonic acid doesn't exist in the free state. When CO2 dissoves in water an equilibrium is set up between CO2 molecules (possibly solvated) and water on one hand and an aquated proton (a hygrogen ion) and the aquated hydrogencarbonate ion on the other. The concentration of hydrogen ions is too low to react with carbonate ions from the calcium carbonate so no reaction! Or in other words, the pH of "dissolved CO2" is not low enough to allow reaction with carbonate ions. In any acid-base reaction the acid must be acidic enough to react with the base which is why carboxylic acids (like acetic acid) react with hydrogencarbonate but phenols don't.
Hope this helps.