Does Gravity / curved space cause rotation? Does Gravity / curved space cause rotation?
Meaning, if a spaceship is heading not directly toward Earth, but slightly off to one side, and when finally being close to the Earth it falls into earth orbit, does the spaceship continue to point in the same directions as it was when approaching the earth, or does it now rotate at a frequency that is equal to its orbital frequency?
I would expect the spaceship to be pointing in the opposite direction after completing one half of an earth orbit.
 A: I assume there is only Earth and the ship in your system.
For one thing the ship will not fall into orbit just like that. While
falling, it gathers kinetic energy which will allow it to leave Earth
again towards outer space, after its trajectory has been deflected by
Earth gravity. The amount of deflection depends on its speed and how
close it comes to Earth. The shape of the trajectory is that of (half)
a hyperbola
  (source: Wikimedia Commons)
Assuming the ship has a long axis (i.e. it is not a homogenous
sphere, or concentric homogenous spherical shells), it will be subjected to a rotation because of tidal forces.
Tidal forces are simple to understand. The ship is
supposed to be a reasonably rigid and strong solid (no need for a General Products
hull though). The trajectory of the center of mass of the ship is the same as if the
whole ship mass were concentrated there.
However, that is not the case. Some parts of the ship are closer to
Earth than the center of mass, while others are further away. So the
parts that are closer to earth tend to be deflected more than the parts
that are further from earth. Or to see it another way, they will be
more attracted by Earth. This creates a torque that makes the ship
rotate around its center of mass, and tries to align its long axis
with he center of gravitational attraction.
How fast this rotation takes place is dependent on actual figures. 
Note that the ship follows a curved hyperbolic trajectory. So, even
assuming that, at some point, the long axis is right on (the tangent
to) the trajectory, thus balancing the gravitational effects between
both ends of the ship, this will no longer be the case a bit later as
the tangent changes it orientation. The forward part will tend to be a
bit outside the trajectory while the rear part will tend to be inside.
This will then be accentuated by the tidal effect so that the forward
part will tend to point more and more outside of the trajectory and away from the Earth which is inside, while the
rear part will tend to point closer to the Earth center of gravity.
However, after the ship passes closest to the Earth, the tidal force
pulling more the rear part than the forward part will tend to realign
the ship on its trajectory, with the forward part in front.
I think the total effect depends on actual figures. I suspect there
may be a residual rotation speed rotating the ship as it departs from
Earth, but I have not done any calculation to be sure of it. This would imply a minute, very very ... very minute, change in Earth
 rotation, because of angular momentum preservation.
Side remark:
If Earth is replaced by a neutron star, the intense gravity
differential between fore and aft parts may also tear the ship to pieces.
A: "When finally being close to the earth it falls into earth orbit"  Won't work.  If the body was gravitationally unbound incoming, you must dump binding energy (do a burn or collide) or it is equally unbound outgoing (OK - Shapiro delay and such.  Close enough).
If you want gravitation theory to be predictive, it operates in spacetime.  What you perceive as curves in 3-space are minimum action straight lines in 4-space.  Linear and angular momenta are conserved (close enough).
The Earth's gravitational field is divergent.  If an orbiting body has a longest principle moment of inertia, that axis at equilibrium will point to the Earth's center of mass as it orbits.  ISS FUBAR flying tangent to the surface is madness, hence its forever failing reaction wheels.  Except for a small central volume, there is no "weightlessness" in ISS FUBAR. 
A: I have some troubles with the other answers. As a sanity check I provide this answer.

I assume that the spaceship is not too big.


*

*If the spaceship enters its orbit initially not rotating (perhaps due to some burn to get into orbit), then gravity will not make it rotate. If it is initially pointing forward relative to its local orbital path, then after half an orbit it will point backward relative to its local orbital path there. (It will keep pointing at the same faraway fixed point, notwithstanding the direction of its actual movement.)

*If it is initially rotating, then it will keep that rotation.

*If it is initially rotating (pitch-downwards) with the orbital frequency, then it will also keep that rotation. Only then it will maintain its pitch relative to the Earth's surface. Only in this case (described by John Rennie, v1) will it appear as though gravity does the rotating, but 1 and 2 make clear that it is not gravity that explains the rotation, but rather the initial angular momentum of the spaceship that is preserved.
The key thing is that the angular momentum of the spaceship, whatever it is, will be preserved. (I think this is what Uncle Al, v1,  means, but I'm not sure.)  And even if it is affected, say because of the weight distribution of the spaceship, or due to relativity considerations, it will not be affected such that a stable rotation would be achieved instantly.
The rotation (in the pitch coordinate) of the spaceship may be adjusted by a reaction wheel. This will adjust the rotation, but keep angular momentum preserved.
A: The same side of the moon always faces the earth due to Tidal Locking, so I think that yes gravity does cause an object to rotate.
A: Here's how I picture this, w/out any calculations. Imagine the spaceship consisting of two massive objects, "front" and "rear" ones, connected with a long massless rod. They follow the same trajectory before and after interaction with the planet, with the "rear" one always a little behind the "front" one. The forces transmitted through the rod are always aligned with the motion: when "front" is closer to the planet and tries to accelerate faster it pulls the "rear", and vice versa. Therefore after the spaceship continues on its (supposedly hyperbolic) orbit the rod would rotate by the same angle the spaceship direction turned.
