To answer this, let's look at the next few billion/trillion/quadrillion/? years, and get a sense of the size of our galaxy and its central black hole.
The first thing that happens related to your question, is that our galaxy and Andromeda collide and merge. This happens in some billions of years. When galaxies merge, the combined galaxy exists, but may have a different form, merged central black holes, and stars (or in some cases even one or both black holes) may be flung out of the combined galaxy. But the galaxy will endure, in some form or other.
That's because a galaxy isn't held together by its central black hole.
A sense of scale: mass
In our galaxy, the central BH has a mass of about 4 -4.5 million suns.
A bigger part is the stars, gas, and other ordinary baryonic matter (some hundreds of billions of stars, although many are red dwarfs and smaller than our sun). The ordinary matter is estimated to be about 600 billion suns,or about 150,000 times the mass of the central black hole.
But the biggest part is dark matter. Explained simply, even taking into account all of the above mass, the galaxy still wouldn't be massive enough to rotate as it does. Calculations show that about 85% of all matter in our galaxy is "dark matter" - a type of matter that isn't made of ordinary atoms, but is suspected to be made of particles that can't interact much except through gravity (so we can't detect it through radiation, it doesn't form planets, stars or black holes, etc). Dark matter would be about 3.5 trillion suns, or about 850,000 times the mass of the central BH.
So the total mass (ordinary + dark matter) is about 4 trillion suns or about a million times the mass of the central black hole.
A sense of scale: diameter
Considering size rather than mass, the central BH is perhaps the size of Uranus' orbit (about 12 light hours diameter).
The visible galaxy is about 100,000 light years diameter, or about 70 million times the BH size.
The extent of the dark matter halo is less certain (and has less of a defined edge), but depending on which research is right, may extend between 500,000 and 1 million light years diameter, or something along those lines (from memory), or a little under half a billion times the BH size.
The central BH contains about a millionth (0.0001%) of the galaxy's mass, and about 2 billionths (0.0000002%) of its diameter.
So the central black hole is actually, and oddly, almost insignificant in terms of our galaxy's present-day structure. It might have been crucial for the formation of the galaxy, but that was long, long ago. It's not the current reason we rotate, and its not the reason we stay in galactic orbit. If it vanished or was ejected tomorrow, nothing at all would change except for a comparatively few stars in the galactic centre that directly orbit the BH. We're nowhere near there. We are in a spiral arm.
The bottom line is, if the central BH vanished or left our galaxy, we and our descendants wouldn't ever notice, except for a change in X-ray emissions from that region (as detected by radio telescopes), and a few very faint stars in that region moving slightly differently over the millennia. That's all.
But as other answers explain, a black hole takes an immense time to evaporate, so in reality, two things will happen:
On a timescale of billions to trillions of years At some point the merging Milky Way/Andromeda galaxy (or a successor galaxy) will keep, merge or eject its central BH. This event won't be an 'end' to the galaxy or the stars in them, although the combined galaxy probably won't be a spiral shape; merged galaxies are common. The combined galaxy will settle down and things will continue.
On a timescale beyond human comprehension (quadrillions upon quadrillions of years) If our universe still exists in its present structure and the standard model and standard cosmology are about right, the central BH will eventually evaporate. But the galaxy (and all galaxies, and most matter) will have decomposed long, long, long before that can happen.