No, carbonated water CO2 + H2O actually forms a weak carbonic acid H2CO3. I suppose a weak acid could be used to clean some items, such as light rust removal. But it would not make for a very good, all-around solvent for removing "dirt".
Carbon dioxide does not exist in liquid form at atmospheric pressure at any temperature. The pressure-temperature phase diagram of CO2 shows that liquid carbon dioxide at 20°C requires a pressure of 30 atmospheres. The lowest pressure at which liquid CO2 exists is at the triple point, namely 5.11 atm at –56.6°C. The high pressures needed for liquid CO2 require specialized washing machines.Source
The link I provided earlier about the Pitt discovery looks promising, but I don't know how close it is to commercialization.