Can this feat be performed without invoking magic? Yes and no.
Sufficiently Advanced Technology is Indistinguishable from Magic:
If you showed a demonstration of Quantum Levitation to a medieval human, they would probably conclude that it was magic because they have no frame of reference for how such a thing could be possible within the realm of physical law.
Except the alien's feat from the question amplifies the problem incalculably: a medieval human is still human, and possessed of roughly equivalent intelligence to a modern human--lacking only education and experience.
Conversely, the aliens may be able to individually comprehend principles beyond even the cumulative capabilities of all humans throughout history.
So instantly moving an entire galaxy would appear to be magical to most modern humans. But, if the aliens wanted to demonstrate their technical prowess, rather than claim any kind of supernatural ability, then they've got two avenues:
1. They have power to spare: Moving that much matte would be incomprehensibly costly, especially if a "brute force" method is used (ie, scanning, destroying, and re-creating every particle in the entire galaxy). The aliens could explain that their energy generation technology has advanced to the point where virtually any task is possible.
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Size matters not: Alternatively, the aliens could demonstrate that their understanding of physics is so great, that they can essentially "edit the source code" of reality. In such a situation, moving a galaxy would be as trivial as moving a blade of grass.
The crucial aspect of the demonstration: If the aliens moved the entire Milky Way so far away that Earth could see no stars outside the galaxy, they would have to omit one crucial thing: light.
If the aliens did the galactic equivalent of a "drag and drop" on the entire galaxy, and included all the light that was present at the time, then people on Earth wouldn't notice a difference for hundreds of thousands of years; they'd need to wait until light (or the lack thereof) from outside the galaxy reached Earth.
However, if the aliens filter out all light when they perform the teleportation (which would be doable if they used the "source code" method above), then they'd get one hell of a demonstration. From the perspective of people on Earth, the sky would go black (light generated from terrestrial sources would return too quickly for anyone to see the interruption)...and the Sun would appear after about five minutes, when its light reached Earth.
The night sky would be utterly devoid of stars for four years, until Alpha Centauri would "appear" as its light reached Earth. Over the years, more and more stars would appear to "wink on" as their light reached us.
That would probably be a pretty effective demonstration of technological might. 99