A loud fill valve would cause the violent shaking behind the fridge?
Guess you need to read more on water hammer.
As others have pointed out, that's a classic water-hammer sound.
The answer to your question above is, "Yep." The resonance created by the hammer is vibrating your fridge water line as well as a bunch of other stuff, but it's most obvious to you at the fridge. Eliminate the hammer and you eliminate the fridge problem. It's not the "loudness" of the valve that's the issue, it's what it's causing the piping in your house to do.
Fluid dynamics can be nasty and not-intuitive: I once was involved in a lawsuit regarding one of the better-known buildings in Manhattan. Massive diesel tanks in the basement. Enormous generators on the roof to power emergency systems and a bank's trading floor in the event of an outage, as well as to provide stable power and take load off of ConEd during NYC Hot Summer Day peak usage. Big CAT diesel engines don't burn all the fuel that is sent into them and thus they have a fuel flow in and a fuel flow, surprisingly, out. The "out" returned back to the basement, so what you had was basically a giant loop up 60-some floors by a pump and back down by gravity. Long story short: it appears that when a return valve was closed in the basement to change fuel tanks during system operation, it somehow caused a 60-floor water hammer of diesel fuel, resulting in loud cavitation and resonance that was heard throughout the building and which was sufficient to blow the fuel filter off one of the generator systems and start a gigantic diesel-fed fire that destroyed everything on the roof. Tens of millions in damage, including the logistics involved in rigging all the burned stuff off the roof of a fully-occupied 60-story building and lifting all the replacement stuff back into place.
Young man that you are from your photo, I'm pretty sure you CAN do this solution as a DIY. Installing a $12 Korky 528MP is certainly something that you can do yourself, and it will probably go a long way to fixing your problem. As Jim points out, making sure your water pressure is within proper limits is also something else that you should do. If you need an expansion tank (or to replace an existing expansion tank or Pressure Reducing Valve at your house-to-street water connection), then you might think about calling a plumber.
Places you can buy a Korky 528MP:
http://www.homedepot.com/p/Korky-Qu...MERCH=REC-_-nosearch2_rr-_-NA-_-203145423-_-N
http://www.lowes.com/pd_336988-868-528MPK_0__?Ntt=528mpk&UserSearch=528mpk&productId=3284948&rpp=32
Installation video (if a nice Wisconsin girl can do it, so can you!!):
Korky makes all its stuff in Wisconsin, USA. Nothing hecho en Mexico, like Fluidmaster.