In order to make an econometric model for explaining the evolution of the US Real GDP, does it make sense (from an economic point of view, not from a statistical one) to include CPI (as a proxy for inflation) among the regressors?
You are not very clear on whether your model is uniequational or a system of equations (e.g. a VAR). I will assume the former is your case.
There are several approaches to estimating Real GDP. One approach is via estimating a production function (i.e. a supply side approach). Here, you need information on factors of production, like hours worked, education level, physical capital, etc. It would make little sense to use inflation as a regressor in this approach.
An alternative method is to focus on the demand side, by which you are interested in explaining the determinants of GDP sub-components like consumption, government spending, etc. This approach is in my opinion the most common among specialised institutions like central banks and etc. For example, Atlanta Fed, OECD, NIESR UK, researchers at the Bank of Canada.
The latter type of models are normally very complex, and go beyond the one equation system. As such, some of them do include inflation as one of the variables (both as independent and dependent variable), because inflation is expected to affect real wages and thereby consumption (for example, see the documentation of the Atlanta Fed model here.
Now, if you only have one equation, and are using the demand side approach, I would include inflation as long as other included variables are also nominal (like retail sales). Alternatively, you can redefine every variable as real, and not include inflation. This is a more restrictive scenario though, as you are imposing extra restrictions (which, in fact, you can test).