Here are some back-of-the-envelope calculations regarding the rotations of stars in a galaxy. (
The calculations make some simplistic assumptions and are only intended to explore the order-of-magnitude reasonableness of the ideas for discussion, not to provide a rigorous solution.)
The qualitative model I'm assuming is thus: intergalatic plasma currents twist around each other; this acts on intragalactic currents, making them turn around the centre with the observed velocities; the intragalactic currents are those responsible for pinching down into stars and forming heliospheres; as the currents move, so too do the stars along their lengths. Neutral matter within the stars are gravitationally bound to the ionised matter, thus also moving with the currents. Similarly for planets in orbit around the stars.
So what we are asking is this: if the moving electric current drags the ionised part of the Sun along with it at the observed galactic rotation rates, will the non-ionised parts of the Sun and the solar system be dragged along gravitationally?
Taking our Sun as an example, using standard numbers and basic equations, and assuming that accepted notions of mass and distance are correct, we find the following:
The mass of the solar system is roughly the same as the mass of the sun, so we'll approximate the whole thing as a point of mass (m) 2e30 kg.
The distance to the centre of the galaxy (r) is 2.5e20 m.
The rotational velocity of the solar system around the core (w) is 8e-16 rad/s.
The centripetal force (F = m.r.w^2) keeping the Sun in orbit is therefore 3.2e20 N.
Now, let us assume that the Sun is homogenous (we can argue about that later). The average density is 1.4e3kg/m^3 and the volume is 1.4e27 m^3.
This gives an average centripetal force of 2.3e-7 N per m^3, assuming 100% ionisation.
For a 1 m^3 region of average density at the surface of the Sun, this centripetal force would equate to an acceleration (F = ma) of 2.3e-7 / 1.4e3 = 1.6e-10 m/s^2. The surface gravity is 274 m/s^2, so the neutral particles should remain gravitationally bound to the accelerating ions. (While the calculation assumed 100% ionisation, the margin is so high here that we should be able to have very little ionisation to ensure an electromagnetically dominated rotation around the galaxy.)
Thus it would seem that the self-gravity of the Sun should be sufficient to hold it together in the presence of an externally applied centripetal force.
Next let us consider the orbits of the planets and, assuming a worst-case of being uncharged and only gravitationally bound to the Sun, is gravity sufficient to keep the planets in orbit when in opposition/conjunction with the galactic core?
The centripetal acceleration (a = r.w^2) towards the galactic core is 1.6e-10 m/s^2. (This is the same number we arrived at above via calculation of the forces, which is a handy consistency check.)
Taking Pluto as the most problematic planet, its centripetal acceleration towards the Sun is 3.8e-6 m/s^2 - more than dominant. Thus the planets would remain gravitationally bound to the Sun and follow it around the galaxy.
The biggest (worst) assumption here is the averaging of the density and the equal distribution of the centripetal and self-gravitating forces. It's not likely to be so simple. Before we can depart from the given assumption, we need to decide qualitatively what the interior structure of the Sun may be and then look to see if it would still hold together, but there would appear to be many orders of magnitude of slack to take up before things fall apart. The tenuous atmosphere of the Sun, being very ionised, is clearly confined to the Sun by local fields (as observed, not just theory), so it is only the internal bulk that we need to worry about.
There are still open questions about the characteristics of the currents, fields, ionisation, and solar structure. What do these need to be to exert a 3.2e20 N force on the Sun?

In the scheme of things, that force would appear to be very small. What are the arguments against it being unachievable?
If there are any mistakes in the maths or reasoning, feel free to point them out.
