You just need to observe the relative carbon isotope levels in the CO2 of the atmosphere. If the CO2 is natural, there'll be a large proportion of C-14 (it's created by interactions with cosmic rays in the upper atmosphere). If it's man made, there'll be a lot of C-12 and 13, since that comes from oil, which has been nowhere near the upper atmosphere for millions of years.
Oh, and there's now enough man-made CO2 in the air that carbon dating is inaccurate for recent things (it tests as really old).
The models certainly do enter into it if you want to predict what the consequences will be of having higher CO2 concentrations. Knowing that we belch a lot of carbon dioxide into the atmosphere may tell us the cause, but not the effect. That has to be done with modeling or educated guesses.
I'll ask you again: without using models, how do we know what the earth's temperature would have been without anthropogenic CO2?
Without knowing that, we can not draw any conclusions as to whether the modern warming trend is caused by human-emitted CO2 or not.
Incidentally, I devoted precisely one line of my post to name-calling, namely reversing a common ad-hominem levied against skeptics of climate modelling.
Increased CO2 is the only reasonable forcing so far; water vapour, methane, etc. have too short a half life in the atmosphere. The bulk of the CO2 increase is anthropogenic in origin, ergo the warming is caused by humans - no models necessary.
How does measuring CO2 levels answer the question of "what would have happened if there were less CO2 in the atmosphere?"
Can we do the same thing with sulfur dioxide after a volcano? If so, all we need to do is wait for the next big volcano to prove geoengineering is viable.
You just need to observe the relative carbon isotope levels in the CO2 of the atmosphere. If the CO2 is natural, there'll be a large proportion of C-14 (it's created by interactions with cosmic rays in the upper atmosphere). If it's man made, there'll be a lot of C-12 and 13, since that comes from oil, which has been nowhere near the upper atmosphere for millions of years.
Oh, and there's now enough man-made CO2 in the air that carbon dating is inaccurate for recent things (it tests as really old).