As I read the question it was more about
H2O + CO2 <-> H2CO3
than about
H2CO3 <-> H+ + HCO3-
Thing is, reaction that we observe - and equilirbium constant of which we can easily measure - is
H2O + CO2 <-> H+ + HCO3-
so in reality we don't know what is concentration of the carbonic acid. I have seen different equilibrium constants reported for both processes.
Quantum calculations show that isolated molecule of the carbonic acid should be quite stable, but its decomposition in the presence of water is very fast. As decomposition produces water, this is an autocatalytic process, hence it is not possible to isolate pure carbonic acid, also concentration of the carbonic acid in the solution is probably very low.
Jae: whether carbon dioxide starts to evolve depends on the concentration of reagents and concentration of produced gas. In more concentrated solutions you will see evolving gas, in diluted it doesn't start to evolve.