How is the impulse-response function of a given system related to the autocorrelation function? If I have the autocorrelation function of an observed system output, how does this relate to the impulse-response function of that system, if I don't have information about the input?
 A: 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 $R_x(\tau) = K\delta(\tau)$ 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.
A: This is an answer from the context of control systems engineering (CSE).  If your vocabulary means something different than what it does in CSE then the following answer might not be as useful.
I'm writing it out, because @Dilip_Sarwate suggests it is relevant as an answer.
The following is often applied to "Linear Time-invariant" (LTI) systems, but because of the instantaneous nature of the Dirac-delta, it can apply with linearization to any system that can be linearized, or to non-linear, non time-invariant systems as long as the derivative of the transfer function exists.  Nearly all real-world physics is governed by the 5 conservation laws (energy, mass, linear momentum, angular moment, and entropy) which can all be expressed in derivative therms.
I'm assuming you don't need me to show the relationship between the Laplace transform and the convolution operation.  If that is not the case, let me know.
The impulse response is defined as the response of a system to a Dirac-delta input, at least in control system engineering.  It is contrasted against the step-response which is the integral of the Dirac-delta over time.  The step response is less directly non-physical, and can often be physically approximated.  To a person in CSE the "I don't know the input" and "impulse response" are incompatible.
Definition of convolution(1):
$$ \left[ f *g \right] = \int_{0}^{t} f\left( \tau \right) g\left(t-\tau \right) d \tau $$
property of integral of dirac delta shown in (2) and (3) of (2)
$$\int_{-\infty }^{\infty } f\left( x \right) \delta\left(x-a \right) dx = f \left( a \right) $$
and
$$\int_{-a - \varepsilon }^{a+\varepsilon } f\left( x \right) \delta\left(t-a \right) dx = f \left( a \right) $$
where $\delta$ is the Dirac delta function.
If we convolute $f$ with a Dirac-delta, we get:
$$ \left[ f *\delta \right] = \int_{0}^{t} f\left( \tau \right) \delta\left(t-\tau \right) d \tau $$
which upon substitution evaluates to:
$$\int_{0 }^{t } f\left( \tau \right) \delta\left(\tau - t \right) d\tau = f \left( t \right) $$
If you give a differentiable system $f$ a Dirac-delta input $g$, and if you can adequately measure the system response (measurability, sampling, discretization are sufficiently capable), then you are directly measuring the system transfer function as defined in CSE.
Here are some extra references. (3, 4, 5)
PS: Dr. Ron Adrian covered this in one of his lectures during the Fluid Mechanics Measurement courseMAE 504.  I would like to think I am presenting it well, but I am quite sure there is room for improvement.
A: The problem is unsolvable.  
Suppose that the autocorrelation of the output of the system is a delta function centered at 0.  This can happen if the output of the system is white noise.  Now, you can imagine infinitely many ways of creating this white noise output with different combinations of inputs and impulse response of the system.  For example, you could use any variety of colored noise as the input and then have the system filter the noise back to whiteness.  
