Sort of, the data looks like CO2 has done most of what it is capable of.
In Theory each doubling of CO2 will cause an added energy imbalance of 3.71 W m-2.
Since about 2001 (when we got satellites up to measure the imbalance) the energy imbalance is still
increasing, but in the short wavelength spectrum. In the longwave spectrum (where CO2 would have a effect),
the energy imbalance is decreasing.
Surface Irradiances of Edition 4.0 Clouds and the Earth’s Radiant Energy System (CERES) Energy Balanced and Filled (EBAF) Data Product
CO2 is a greenhouse gas because as you add more the positive energy imbalance continues to increase.
Each unit of increase has less forcing than the unit that came before, because as Earth warms, it radiates better.
Where this would break down is where when CO2 is added the increased imbalance in the longwave radiation is exceeded
by the increase in outgoing longwave radiation, which looks like it is happening.