The autocorrelation function $R_y$ of the system output $y(t)$ is related to the autocorrelation function $R_h$ of the system impulse response $h(t)$ and the autocorrelation function $R_x$ of the system input $x(t)$ as $$R_y = R_h star R_x tag{1}$$ where $star$ denotes the convolution operation. But note that because $R_h$ and $R_x$ are (real-valued) even functions of time with a peak at the origin (as indeed is $R_y$), we can also think of the result as saying that $R_y$ is the cross-correlation function of $R_h$ and $R_x$. Unfortunately, all this does not help you very much if all you know is $R_y$ and have no idea what $R_x$ is (or possibly even what $x(t)$ might have been). If $x(t)$ is indeed white noise with autocorrelation function $R_x(tau) = Kdelta(tau)$ where $delta(cdot)$ is the Dirac impulse, then you can estimate $R_h(tau) = K^{-1}R_y(tau)$ which is progress of sorts, but it is still not possible to say exactly what the impulse response $h(t)$ might be. Quite different signals can have the same autocorrelation function: that is, knowledge of $R_h(tau)$ does not tell us what $h(t)$ is, though it does reduce the possibilities considerably.

The autocorrelation function $R_y$ of the system output $y(t)$ is related to the autocorrelation function $R_h$ of the system impulse response $h(t)$ and the autocorrelation function $R_x$ of the system input $x(t)$ as $$R_y = R_h star R_x tag{1}$$ where $star$ denotes the convolution operation. But note that because $R_h$ and $R_x$ are (real-valued) even functions of time with a peak at the origin (as indeed is $R_y$), we can also think of the result as saying that $R_y$ is the cross-correlation function of $R_h$ and $R_x$. Unfortunately, all this does not help you very much if all you know is $R_y$ and have no idea what $R_x$ is (or possibly even what $x(t)$ might have been). If $x(t)$ is indeed white noise with autocorrelation function $R_x(tau) = Kdelta(tau)$ where $delta(cdot)$ is the Dirac impulse, then you can estimate $R_h(tau) = K^{-1}R_y(tau)$ which is progress of sorts, but it is still not possible to say exactly what the impulse response $h(t)$ might be. Quite different signals can have the same autocorrelation function: that is, knowledge of $R_h(tau)$ does not tell us what $h(t)$ is, though it does reduce the possibilities considerably.