The maximum positive correlation between ocean temperatures and atmospheric CO2 is within a range from 0.45 to 0.48, depending on which dataset is considered (HadSST2, NCDC or UAH).
With a sample size of 361 (number of monthly DIFF12 values) these correlation coefficients are highly significant at the 0.05 level, and correspond to a goodness-of-fit (r2) ranging from 0.20 to 0.23.
This represents a fair degree of explanation, and far bigger than achieved by any other factor considered in the present analysis, but it also suggests that there are other factors beyond ocean surface temperature which have influenced observed changes in atmospheric CO2 since January 1980.
Examples of such potential factors are changes in soil moisture, living biomass, volcanic eruptions, geological weathering processes, burning of fossil fuels, etc.
The correlation between CO2 released by anthropogene sources and changes in atmospheric CO2 is not stable (Fig. 15), and not able to explain much of the observed increase in atmospheric CO2 since January 1980. A qualitatively identical conclusion may possibly be suggested for the effect of volcanic eruptions during the study period, but the character of the volcanic data available does not make it possible to carry out a comparable statistical analysis on this.
Actually, on the time scale investigated, the net effect of a major volcanic eruption appears to be a reduction of the prevailing increase rate of atmospheric CO2, probably an effect of ocean cooling induced from cloud effects.
http://www.sciencedirect.com/science/article/pii/S0921818112001658