The source you mentioned tries to explain why CO2 lags by about 1000 years in historical records.
Henry's law tells you something about the equilibrium between the CO2 level in the surface layer and the atmosphere. However, this is not the reason for the 1000 year time-lag mentioned. There's not enough CO2 in the surface layer to cause these increases in atmospheric concentration. What is important is the exchange of CO2 between the deep sea and the surface layer. This seems to have a typical timespan of 1000 a.
Citing from your source:
As the Southern Ocean warms, the solubility of CO2 in water falls (Martin 2005). This causes the oceans to give up more CO2, emitting it into the atmosphere. The exact mechanism of how the deep ocean gives up its CO2 is not fully understood but believed to be related to vertical ocean mixing (Toggweiler 1999). The process takes around 800 to 1000 years, so CO2 levels are observed to rise around 1000 years after the initial warming (Monnin 2001, Mudelsee 2001).
So the initial increase in CO2 levels due to Henry's law trigger via feedback mechanisms changes in the vertical circulation pattern thereby transporting more CO2 from the deep sea to the surface. It's this time constant that is seen in the records.