Carbon dioxide does dissolve in water.
Carbon dioxide dissolves in water to form an aqueous solution. This may seem counterintuitive. Carbon dioxide is a gas and water is a liquid. However, gases, liquids and even solids are all chemicals. This means they can all dissolve in solvents. Dissolution creates a homogeneous solution with components derived from solutes and solvents. In this case, water serves as the solvent and carbon dioxide serves as the solute.  The reaction is a favorable one, which produces carbonic acid. 
Like Dissolves Like: A Rule of Thumb
A basic rule in chemistry is “like dissolves like.” This is often interpreted to mean solute-solute reactions and solvent-solvent interactions occur first. These reactions must be broken before solute-solvent interactions can take place. This necessitates energy. Afterwards, solute-solvent interactions can develop and release energy. If the released energy is greater or equal to the amount required to break solute-solute and solvent-solvent interactions, then solute-solvent interactions will occur spontaneously. According to this rule, carbon dioxide is not likely to dissolve in water. 
Entropy: The Unspoken Component
“Like dissolves like” is a rule of thumb, but it is not the complete story. Entropy is also involved. In situations where pressure and temperature can change, entropy can and does drive reactions even though these interactions seem to contradict “like dissolves like.” 
Carbon Dioxide in Water: Amounts May Vary
The amount of carbon dioxide that dissolves in water is dependent upon several factors. These include temperature, pressure and pH.  Reactions attempt to achieve equilibrium. In this case, equilibrium is not a concentration quantification, it is an equalization between the forward and reverse reaction rates. When the reaction achieves equilibrium, the concentrations of carbon dioxide, water, and carbonic acid can be related by a mathematically derived ratio known as the equilibrium constant. However, actual amounts will need to be assayed. 
 OSU Department of Chemistry
Chem 1125 Chapter 15
 Encyclopedia Britannica
Carbonic Acids and Carbonic Salts
US Department of Energy- Ask a Scientist
Dissolving Carbon Dioxide into Water
 Utah State University
Dissolved Carbon Dioxide
 University of California Davis-Physical Chemistry
The Equilibrium Constant
Glossary of Terms
Aqueous Solution: a solution in which water is the dissolving medium or solvent.
Dissolution: the process by which a solute forms a solution in a solvent.
Solvent: the component whose phase is retained when the solution forms; if all components are the same phase, the one in the greatest amount is the solvent. If one of the components is water, water is always the solvent.
Oklahoma State University
Solute: dissolved substance; especially : a component of a solution present in smaller amount than the solvent.
“The maximum total amount of CO2 that may dissolve in water is a function of pH. This is because as pH changes, the fraction of the total dissolved Co2 as Co2 (aq) also changes.”
Dissolved Carbon Dioxide Utah State University