Electrons accelerated to several kV energy and impacting a heavy metal target do generate X-rays. This can't happen in an atmospheric pressure discharge, because free path length of charge carriers is too small to get considerably accelerated. An electric discharge produces UV radiation and toxic gases, but the effect can be almost neglected at low currents.
P.S.:
As an update, it's not completely impossible to generate x-ray in atmospheric pressure dsicharges. They can be observed particularly under "exotic" conditions like fast pulsed discharge with sufficient energy. Rather unlikely in a regular spark gap.