A doubling of CO2 will produce approximately a radiative forcing of 3.7W/m^2 as measured from the tropopause and a resultant tendency to warm the surface a bit less than 1.2C at equilibrium in a hypothetical situation with zero feedbacks. This follows rather straightforwardly from basic physics.
However, in the real world there will be feedbacks within the climate system in response to that initial 1.2C of warming per doubling of CO2. Cloud and aerosol feedback, water vapor feedback, lapse rate feedback, carbon cycle feedback and ice albedo feedback are the most prominent examples. Some of these give positive feedback, some negative feedback and all to varying degrees of impact. There are very complicated interactions involved which are in some cases not well understood at all. Climate science refers to these feedback processes and interactions as the climate’s sensitivity to an initial perturbation.
Equilibrium climate sensitivity is not an unknown. It is not exactly pinned down to much better than a factor of 3 however. So we have quite a bit of uncertainty involved. There likely is no rock hard figure, each set of particular global configurations probably has its own unique climate sensitivity to an initial warming influence the equal of 1.2C.
However, by assessing past climate change (the past several ice age cycles), volcanic eruptions and yes, computer modeling we have honed in on a most likely range of sensitivity ( 2C – 4.5C).
Here is a paper describing a determination of climate sensitivity which does not involve modeling:
Using NCEP reanalysis data that span four and a half solar cycles, we have obtained the spatial pattern over the globe which best separates the solar-max years from the solar-min years, and established that this coherent global pattern is statistically significant using a Monte-Carlo test. The pattern shows a global warming of the Earth’s surface of about 0.2 °K, with larger warming over the polar regions than over the tropics, and larger over continents than over the oceans. It is also established that the global warming of the surface is related to the 11-year solar cycle, in particular to its TSI, at over 95% confidence level. Since the solar-forcing variability has been measured by satellites, we therefore now know both the forcing and the response (assuming cause and effect). This information is then used to deduce the climate sensitivity. Since the equilibrium response should be larger than the periodic response measured, the periodic solar-cycle response measurements yields a lower bound on the equilibrium climate sensitivity that is equivalent to a global warming of 2.3 °K at doubled CO2. A 95% confidence interval is estimated to be 2.3-4.1 °K. This range is established independent of models.