Your comparison to water vapor is a bad one.
The amount of water vapor in the atmosphere does increase with atmospheric temperature. This is because more evaporation occurs and can be held as vapor longer, thus offsetting the equilibrium of water vapor. Cold air will cause more water vapor to condense back to liquid. This process of water cycling occurs on the order of hours/days.
This is in stark contrast to carbon dioxide, which does not condense out of the atmosphere when it cools. Nor does carbon dioxide evaporate from the surface of the earth. Instead, it is a byproduct of respiration and combustion. Carbon dioxide stays in the atmosphere for 100s of years before it gets deposited or chemically converted. Furthermore, carbon dioxide becomes part of geological processes in the carbon cycle that takes 1000s of years, not days.
So, to summarize, carbon dioxide does not directly increase in the atmosphere due to temperature rise, because it is not constantly evaporating from the Earth's surface and condensing in the atmosphere. Carbon dioxide does not condense at normal atmospheric temperature... it won't condense until it gets as cold as -78.5 degrees C, which does not occur on the surface of the Earth [in this climate regime].
It should be noted that carbon dioxide will indirectly increase in the atmosphere with temperature rise. This is because of feedback mechanisms. For instance, with increasing temperature, trees will be more dry on average and will thus combust more easily. More wildfires are expected as temperatures rise, and with them come more carbon dioxide emitted during the fire. There is also the possibility that increases in global temperature will cause feedbacks in the geologic cycle that lead to more volcanic activity, which is another source of carbon dioxide emissions. And, as Michael points out, the methane released when permafrost melts will eventually oxidize with the hydroxyl radical into carbon dioxide and water vapor.