This answer and the excellent article Gravity Gradient Stabilization of Earth Satellites explain that you need to use a damper of some kind, where friction or other lossy processes slowly absorb rotational energy, like a lossy pendulum slowing down.
One caveat is that in the the final state, the spacecraft is actually rotating once every time it revolves around the Earth, so that one face of the spacecraft is always pointing in the nadir direction.
See also this answer and the extensive series of comments be low it.
This can be done for example with a weight on a boom with a spring and dashpot, or with the flow of viscous liquids in curved tubes
two purposes that the gravity gradient serves are:
- Provide a non-symmetric resorting force so that the spacecrafts rotation rate has a non-uniform or oscillatory component. That causes the damper to be excited back and forth each orbit in order to produce friction.
- Define the final, lowest-energy orientation/attitude state (always pointing down as it orbits) when the damper has finished absorbing rotational energy. This takes on the order of days or weeks but it depends greatly on the details of the design.
The rotational angular momentum of the spacecraft on the other hand, is constantly sloshing (being exchanged) back and forth with the Earth's angular momentum relative to the Earth-spacecraft center of mass the on a timescale of hours, so there are no conservation law quandaries.