A pure inductance only connects electrical current to magnetic field in space.
An antenna must connect electrical voltage and current, to electric and magnetic fields in space.
As real, physical inductors have nonzero size, they do exhibit some electric field. It's just not the focus of the component, and therefore they typically make terrible antennas.
There is a tradeoff between antenna size, bandwidth and directionality. Basically, you can take a suitable inductor, tune it to a particular frequency, and have it resonate with a humongous voltage. (The excessive voltage is necessary to couple effectively with the radiating electric field.) This gives an electrically small mono/dipole antenna, with extremely narrow bandwidth, and acceptable efficiency.
Remember that you can always make an antenna worse than ideal (ideal, given its size, bandwidth and directionality, that is), but you can never make one better. For example, compare an air-core solenoid winding (like a Tesla coil secondary) to a shielded, ferrite cored inductor: the former can radiate effectively if made to resonate with low losses and high Q factor; the former will never achieve more than pitiful efficiency.
An identical argument applies to capacitors, of course; in that case, a practical example is the patch antenna, where a small parallel-plate capacitor is made to resonate at a certain frequency. The bandwidth is small, but that might happen to be okay (for example, a Bluetooth antenna needs hardly 1% bandwidth, so can be smaller than a fingernail).
Tim