Is it possible to create an external electric field using a toroid?
Firstly the magnetic field described in the question

In a solenoid yes, the magnetic field comes out (and back in) at the ends of the solenoid but with a toroid (which doesn't have ends by definition), there can be no magnetic field exiting theoretically. Of course there are small amounts of flux that are still measurable close by the toroid but these are not significant.


  Is this possible to create a reasonably strong electric field with a
  toroid? (Stronger than say, the E field from just current flowing
  through a wire).


The strong E field usually associated with (say) a quarter wave dipole antenna is due to standing waves making the tip of the antenna produce large "resonant like" voltages. This can generate a huge electric field and will, due to the antenna capacitance to its opposite end (ground on a quarter wave dipole), can produce significant currents that give rise to a magnetic field hence the beginning of the creation of an EM wave.

To answer your question I don't see how strong electric fields are generated with a toroid compared to a regular antenna.

At the end of the above quote, you say: -


  E field from just current flowing through a wire


An E field is due to the voltage not the current.