Is there any connection between frequency response of a system and spectrum of a signal? For instance, an AR process can be generated through a white noise $w[n]$ passing through an all-poles system (IIR), 
$$y[n] = \sum_{k = 1}^{p}a_ky[n-k] + w[n]$$
and we can analyze the IIR system, instead of obtaining the spectrum of the signal, to get some properties of the signal and therefore apply it to some applications, e.g., prediction. 

My question is whether all the signals, stationary or non-stationary, can be treated as it is generated via some system driven by white noise, and we can analyze the system, which hopefully can completely characterize the signals.
As mentioned in my comment, look at the Wold Decomposition Theorem :

https://en.wikipedia.org/wiki/Wold%27s_theorem

but actually showing how it relates to your question,

http://www.phdeconomics.sssup.it/documents/Lesson11.pdf.

In simpler terms, your AR system can be represented by a MA system (possibly infinite order) and as the second reference shows, an ARMA system.  The process has more than one representation so there isn't a one to one correspondence between the time series and some system that generated it. 

So the answer is yes, there is a correspondence and often the system itself, is of interest, but the time series and system are not uniquely equivalent. 

another way of thinking about it is that $H(\omega)$ has a phase and magnitude response that gives $|H(\omega)|^2$ but there are other possible phase responses that results in the same magnitude squared.