The fine-tuning argument is a species of the argument from design.
First, there are quite a few fundamental constants; and taking Occam's razor onto account, a single universal constant, of which the other constants were mere aspects thereof would be the best possible 'reduction'; then, the natural response would be to set this universal constant to the value 'one'; and then it is as though there is no constant - for what can be more natural than this value?
If this is possible, then is there such a thing as a fine-tuning argument?
Though we can't do this, we can to some extent; in the system of natural units - the speed of light c, plancks constant h and Newtons gravitational constant G are all set to one; that this is a 'natural' thing to do can be seen by the simplification achieved in an equation from Special Relativity:
E^2 = p^2.c^2 + m^2.c^4
Which becomes:
E^2 = p^2 + m^2
Which demonstrates the reappearance of some antique Greek Mathematics - Pythagorases Theorem.
What this suggests is that there is a natural scale of lengths; and taking into account, conjecturally that nothing is continua as such, but fine-grained, atomic or quantised; then we might conjecturally suggest that the natural scale then is the minimal values of such: ie the minimal charge, of which the others are multiples; or the minimal mass and etc, etc.
But then the fine-tuning question disappears; and the question that replaces it is - why is the universe fine-grained or atomic? A better question - physically and philosophically - possibly.
So, does the argument above refute the fine-tuning argument, contradict it or re-orientate it? I would suggest the latter.
And this done without a cornucopia of multiverses being conjured out of wormhole!