What is minimal aperture/magnification to recognize phases of Galilean moons? Given you take a (non-amateur) refractor telescope, what are aperture and magnification required to recognize phases of the four Galilean moons?

What did Galileo see:
http://www.astro.umontreal.ca/~paulchar/grps/site/images/galileo.4.html
From Earth the resolution of is limited by the atmosphere.  Turbulence in the atmosphere make it impossible to resolve the moons of Jupiter to more than a few pixels wide. 

Amateur refractor telescopes tend to be small and less powerful than reflector telescopes. I'm not aware of any amateur refactor telescope that could resolve a moon of Jupiter to more than a point of light. The eye is unable to resolve an object that is less than about 60 arcseconds and it would need to be closer to 1000 arcseconds before many surface details became visible. The Moon, for comparison, is 1800 arcseconds.

Professional equipment can partially overcome the atmosphere by using adaptive optics, but this is not a feature that is generally available to amateurs.

Jupiter and its moons orbit outside the orbit of Earth, so from our point of view Jupiter is never partially lit less than 99%. The moons are sometimes eclipsed by Jupiter, but they too are never partially lit less than 99%.  With amateur equipment, Jupiter and its moons are never anything other than "full".

Even with the Hubble telescope, the resolution is not sufficient to identify phases of the moons.

What Galileo saw was Jupiter, resolved as a disc and 4 moons orbiting it which appeared to him as "stars" (he called them the Medicean stars). The four moons were not always visible, sometimes a moon was hidden in the shadow of Jupiter, or behind Jupiter from our point of view. At other times a moon would be too close to Jupiter to be visible, or outside the field of view of his telescope. 

What he realised was that these were bodies in orbit around Jupiter, and that they obeyed the same rule for orbits that Kepler had proposed for the planets.