From a pedantic point of view it will be impossible to prove something
is gibberish if it's creation involves a random element. You will not
be able to prove that the random element is actually random and contains
no information.
However, if the process that created it was deterministic (like moving
a grill in a consistent pattern over a table of characters), you can (at
least in theory) reconstruct the process and by virtue of proving there
was no choice involved, show that there is no information contained in
the message. Although you would have the further possibility that there
could be information in the underlying 'table of characters' !!!
If the creation process involved arbitrary human choice you will similarly
not be able to prove it is gibberish, although the evidence might be beyond
reasonable doubt.