Q:

Could we use experimental evidence of micro black holes to prove the existence of additional spatial dimensions?

They are planning to create a new particle accelerator that can smash particles together at a much higher energy than the LHC.
If in the process we observe the decay of a micro black hole, would that be proof of additional spatial dimensions?

A:

Extra dimensions are predicted in string theories which are  not measurable in the standard versions, they are compactified :

In compactification, some of the extra dimensions are assumed to "close up" on themselves to form circles. In the limit where these curled up dimensions become very small, one obtains a theory in which spacetime has effectively a lower number of dimensions.

This is the space time we live and experiment in.
Since about 1998 there has been a promise of large extra dimensions to be seen in  particle experiments. I remember this paper and in a few years there were also programs generating the appearance of micro black holes in the future LHC experiments.
CERN is still answering questions about the dangers of black holes in the accelerator eating up the earth: there are none.
The experiments search for the signature of these small black holes, and have only given limits up to now , as you can see if you go to the document server and search for "limits on large extra dimensions" .
So if in a new collider there is a signature for a mini/small black hole, then it will be a validation for models that propose large extra dimensions. The signature would be for a jet of particles showing just thermodynamic behavior and not following the strong interactions which generates the usual jets.

