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As long as neither spacecraft nor the astronaut are accelerating or decelerating, the relative speed of the spacecraft and the astronaut remains the same. So the astronaut will hover near the spacecraft.

The actual velocity is irrelevant here, it's the same principle with every spacewalk: the ISS is moving at about 27,600 km/h, yet the astronauts do not "get left behind" when they exit for a space walk. They, too, move at about 27,600 km/h. They do move at a very slight relative velocity when they move along the spacecraft, though.

Things change if your spacecraft is accelerating or decelerating, though: in this case the astronaut needs to remain attached to the spacecraft to not get lost. As soon as they would let go, their current velocity would remain the same but the spacecraft would continue to change its velocity and the two would get further and further apart.

It turns out that outer space is not a perfect vacuum: there are a few hydrogen atoms per cubic centimeter. (reference)

For large X, non-relativistic physics, the astronaut and spacecraft will stay close enough to each other.

Once X gets small, and you approach the speed of light, these hydrogen atoms could slow down your spacecraft. Therefore, to maintain constant speed against this "apparent headwind" you'd have to apply force to the spacecraft, and the space-walker would not be subjected to that same force.

My hypothesis is that the astronaut will slowly be left behind.

I feel this sort of question benefits from a series of thought experiments.

Imagine instead that you've got two astronauts, side by side, zipping through space at some constant speed.

They're kind of sweet on each other so they're holding hands. Awwwww.

But then they suffer a cruel change of heart and stop holding hands!

What do you imagine would happen?

Does anything change if one of the astronauts is much fatter than the other?

If we replace the very fat astronaut with a spacecraft, does that change anything?

(I'm asking these questions quasi-rhetorically, for the benefit of the original question-asker. No need to answer me in comments.)

Another way to think is to consider two space walking astronauts; one inside the ship and one outside. Neither is touching the ship, both are moving at essentially the same speed in the same direction. All three pretty much stay together.

However, there could be a teeny tiny amount of acceleration experienced by each. For example, at an extremely high velocity, even the tiny impulse caused by each interstellar proton hitting a body can cause a bit of drag. The "indoor" space walker won't experience it, and so won't be slowed at all, but the ship will, and so will the "outdoor" space walker. It's not clear which one would be affected more, it depends on their cross-sectional areas and masses.

Then there are tidal effects. If there is a distant gravitational source, and there always is, that will accelerate all three the same. But if you are fairly close to a source of gravity, then it is possible that it affects them slightly differently because they will each have a very slightly different distance from the source.

For more on that see answers to Lowest ISS microgravity and for fun see How to get sunburned through the window of a General Products hull?

And before your ship does another neutron-star flyby to accelerate so fast, remember that what humans call UV is not the only thing that gets through a General Products Hull!

no, conservation of momentum is retained (an object in motion will remain in motion unless something acts upon it)...similar to being in an airplane and throwing a ball up in the air...seems like it should fly to the back of the airplane, but it won't...it'll act just like you were on the ground.

A brave Astronaut is leaving the spacecraft to a space walk, while not being attached to the spacecraft.

Will the astronaut

hover near the spacecraft at the same speed as it (1/X of speed of light), or

be quickly behind the spacecraft and will watch it disappear in the black horizon?

Newton's First Law of Motion ("an object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force") means that the astronaut -- who is traveling at the same speed and direction as the ship while inside the ship -- will continue traveling at the same speed and direction as the ship when he steps out of it.

The correct and complete answer is distributed among many previous posts. I try to condense them here, without attempting to reference all of you guys. All of the below information was provided in the previous answers.

The main point is that

• 
  neither the ship nor the astronaut tend to brake in empty space because of Newton's law.

Additionally, there are three very weak effects:

1. 
   Space is not completely empty. This depends on where you are, but there will be some countable amount of atoms (mostly hydrogen) per cubic meter. These take away your velocity, very very slowly. Whether the astronaut's or the ship's velocity decreases faster depends on the ratio of their mass to their cross-sectional area, respectively.


2. 
   Tidal effects also pull them apart. This is because they are located at slightly different distances to the surrounding sources of gravity. The closer you are to such a source, the stronger is the respective force, hence the astronaut and the ship experience different gravitational pulls.


3. 
   Mutual gravity pulls them together. Both the spacecraft and the astronaut have mass and hence attract each other.

Whether the astronaut will be able to measure a change of the distance between her or him and the spaceship (during her or his lifetime) depends on the exact initial conditions.

Let's tackle this with a slightly different question:

Which falls faster? A bowling ball or a feather?

Now, everyone knows the feather will fall slowly, but that's because the feather has a massive surface area to catch the air around it. Without air resistance they fall at the same rate (see the video below for a most impressive display of that principle)

If an astronaut exits a spacecraft moving at 17,000 mph, they're still moving at a relative 17,000 mph because there's nothing to slow the astronaut down.

As others have explained, cosmic dust and orbital mechanics aside, the astronaut will cruise along with the ship. However, to make sure we cover all the aces, he'd better check the ship is not rotating before he leaves.

If it is, then while he is inside, he will find himself held to the outer walls by "centrifugal" force (really, it's the walls pushing him round in a circle). Once he exits, that pushing will send him drifting off at a tangent to the rotation. Since the craft will turn under him as he floats away, it will look like he is moving straight out from the door. At this point, a Wilhelm Scream might be appropriate.