System Identification (Linear Passive Circuit)
IF the system is linear, then you are right, it can be characterized solely from its impulse response.  It can be thought of as convolving the input by the impulse response.

Often the impulse response is difficult to meaure.  One way around this is to measure the step response, which is usually easier, then differentiate that.  That may have a lot of noise, but after some low pass filtering you can often get a reasonable impluse response that is impossible to measure directly.  I go into a lot more detail, including a real world example, here: https://electronics.stackexchange.com/a/27861/4512