Could galactic rotation be similar to an irrotational vortex? Could black holes' near light speed rotation cause galaxies to move like an irrotational vortices? 
 A: If we take the Milky Way as an example, the black hole at the centre, Sagittarius A$^*$, has a mass of about 4 million times the Sun. However the mass of the Milky Way is somewhere around a trillion Suns. So the central black hole makes up 0.0004% of the total mass. While the central black hole may have been important in the formation of the Milky Way, its effect on the overall dynamics is negligable.
In any case, the rotation curves of galaxies are almost flat and do not in the least resemble an irrotational vortex.
A: Comment to the question (v1): Besides what John Rennie wrote in his correct answer, note that the velocity profile of an irrotational/free vortex falls off as 
$$\tag{1} v~\propto~ 1/r,$$ 
while a galactic rotation curve may actually increase with $r$, and in any case, it never falls off faster than$^1$ 
$$\tag{2} v~\propto ~1/\sqrt{r},$$ 
so the analogy mentioned in the title has no merit whatsoever. 
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$^1$ The lower limit (2) corresponds to Newton gravity without dark matter and all mass concentrated in the center.
