Actually, while a small amount of the CO2 does bond with the water to form carbonic acid. (H2CO3) The carbonation is based around dissolving CO2 gas into the water, much like dissolving salt into water. Most of the CO2 (I think about 98%, but don't hold me to that figure) remains CO2, just dissolved in water. CO2 is dissolved in water at high pressures and low temperatures, but at atmospheric pressure tends to return to a gaseous state. The soft drink fizzes when you expose it atmospheric pressure as the CO2 energetically bubbles out of solution in the process lifting small particles of the liquid, the soft drink only fizzes visibly for a short time as the kinetics of the change in state slow as the concentration of carbon dioxide decreases. Overall the carbonic acid reaction does little in the bulk of the process, the concentrations of carbonic acid are quite low, a low equilibrium constant for the reaction.
If you replace the lid on the container the CO2 continues to bubble out until it reaches an equilibrium pressure with the CO2 trapped in the container. If allowed to sit the soft drink goes flat as the CO2 that was dissolved into the water returns to gas until the point at which the amount of CO2 remaining in the drink is equivalent to how much will naturally dissolve into water at atmospheric pressure.
Interestingly enough the ability for gases to dissolve into liquids is the basis for all fish life, the oxygen that dissolves into the water is what gills remove for the fish to live on. This also explains why thermal pollution of rivers and lakes has such a bad effect on aquatic life, the concentration of O2 dissolved in water is very temperature dependent, and the higher the temperature the less oxygen available.