In this work we consider the most general electromagnetic theory in curved space-time leading to linear second order differential equations, including non-minimal couplings to the space-time curvature. We assume the presence of a temporal electromagnetic background whose energy density plays the role of dark energy, as has been recently suggested. Imposing the consistency of the theory in the weak-field limit, we show that it reduces to standard electromagnetism in the presence of an effective electromagnetic current which is generated by the momentum density of the matter/energy distribution, even for neutral sources. This implies that in the presence of dark energy, the motion of large-scale structures generates magnetic fields. Estimates of the present amplitude of the generated seed fields for typical spiral galaxies could reach 10-9 G without any amplification. In the case of compact rotating objects, the theory predicts their magnetic moments to be related to their angular momenta in the way suggested by the so called Schuster-Blackett conjecture.