Would an object orbiting another due to gravity start to rotate? I am coding up another JavaScript simulation with a spaceship orbiting a black hole (kind of like this game). I am wondering if the spaceship went around the black hole if it would begin to spin, or if its angular momentum would stay constant.
 A: Both comments have good points that are two sides to the same coin. Tidal forces are due to differential force on a object. For example, the force of gravity will be different at the head and tail of a ship. Depending on the orientation of the ship this can cause a net torque and then rotation. But the effect is going to depend on how different the gravitational force is from end to end. It is going to be smaller for smaller objects and may not show up at all for something like the ISS in orbit around Earth (Wikipedia: Tidal locking Timescales). But yes to a long enough ship, close enough to a black hole. 
This effect will cause rotation but it won't just spin any which way. If fully  tidally locked then the ship will revolve once for each rotation. It could also oscillate around the tidally locked orientation. It is also possible for the ship to freely revolve while tidal forces cause it to speed up and slow down periodically. 
According to Wikipedia: Tidal locking Timescales, the time for a body to become tidally locked is
$$
t_\text{lock} \approx \frac{\omega a^6 I Q}{3 G m_p^2 k_2 R^5 }
$$
where $R$ is the mean radius of the satellite, $m_p$ is the mass of the planet (or in your case black hole), $a$ is the semi-major axis of the satellite's orbit, and $\omega$ is the satellite's initial spin rate. Follow the link for the definition of the other variables. The key thing to take note of is that the tidal locking time is a strong function the size of the satellite $R$.  Small $R$ means big $t_\text{lock}$.  
