In principle, yes, this is possible. However, it's important to note that this requires an interaction that goes beyond the Earth's magnetic field - you need to harness the gradient of this magnetic field. This is very weak at the length scales available to a satellite, so you would really struggle to make this work, but in principle the effect is there.
A bit more concretely, if you set up a current loop around the satellite with magnetic dipole moment $\mathbf m$, then the satellite would experience two separate effects:
- a torque $\boldsymbol{\tau}=\mathbf m \times \mathbf B$ in the presence of a magnetic field $\mathbf B$, and
- a force $\mathbf F = \nabla(\mathbf m \cdot\mathbf B)$ given by the gradient of $\mathbf B$.
The first effect is real, and it will dominate the interaction; indeed, it is actively used in spacecraft for attitude control as an alternative to reaction wheels or other such mechanical devices. These magnetic alternatives are known as magnetorquers, and they do work in low Earth orbit (but they become less effective at higher altitudes).
To put in some numbers, suppose that you set up a current loop of $1\:\mathrm A$ with an area of $1\:\mathrm m^2$, adding in a factor of $100$ to account for multiple loops, and a possible ferrite core. Using the Earth's magnetic field strength at the surface at about ${\sim}50\:\mu\mathrm T$ for simplicity, you get a torque on the order of $\tau\sim0.005\mathrm{\:kg \:m^2\:s^{-2}}$; this is not a lot, but you can see that it can be improved on to give a meaningful torque, particularly if you can wait and let this act over multiple minutes.
Similarly, you can see how this becomes much less useful as a force. The magnetic field strength, $B\sim50\:\mu\mathrm T$, is already relatively weak, but the gradient is very, very small, since the magnetic field only changes appreciably on length scales on the order of hundreds or thousands of kilometers. If you put in such an estimate, $\nabla B\sim50\:\mu\mathrm T/100\:\mathrm{km}$, you get a much lower number for the force, on the order of $F\sim50\times10^{-9}\:\mathrm{N}$. This is no longer useful for shuttling around spacecraft in the $M>\mathrm{kg}$ regime. The force is there, it's just not a major player in where the spacecraft ends up.
Finally, just to make sure this doesn't become an issue: if you do use such a system to turn or move your spacecraft, it goes without saying that the energy for doing this comes from whatever is driving the current. If the spacecraft turns or moves to a region with smaller magnetic field, the magnetic flux through the current loop will change, which by Faraday's law requires a nonzero electromotive force to form within the loop; the spacecraft's power source must then expend work against this EMF to keep the current going.
