Disciplinary issues are in two categories for this sort of thing, misconduct and gross misconduct. I would expect this is misconduct for which the sanction, if demonstrated is likely to be a written warning, or perhaps a final written warning.
Is your conduct from what you have said likely to be answerable to misconduct? - as a former HR manager then consultant for many years, I think it is something I would wish to investigate.
Out of interest, does your contract or related T&Cs say anything about not working for another employer?