Sorry for not reading your message for a few days. It's difficult to understand the information from your inputs above, so let's first summarize the scenario.
One of your employees has resigned. Then he sent some company-related information to his personal email ID. The IT department tracked this email and issued him a warning.
The employee has managed to convince management that the data was sent 'unknowingly'. If this fact is agreed upon, then there is no question of action like termination or non-completion of relieving formalities.
It seems that even though the IT infrastructure is present to 'track' these incidents, it does not 'restrict' them. Your company does not seem to have clear policies about information security or IT restrictions. You are saying the data cannot be 'proven' as proprietary to the company as well.
The data collected by employees on the job is company property. Though if the policies do not enforce this, it will not be possible to take any action against such information theft.
My opinion would be to let go of the employee (either through resignation or termination) if he is not of further value to the organization. If you do not want to risk further incidents with this employee, you can consider relieving him sooner than the notice period. Also, it would be wise to immediately restrict his access to 'valuable' data in the company using IT policies.
While relieving the employee, follow "strictly" the terms mentioned in his appointment contract in this regard. Even employees who do not follow general ethics should be treated ethically and legally by the company.
Also, please discuss this matter with your seniors in the company, as they might have different or a deeper understanding of the company policies than you or other employees.
Going forward, you should try to set up policies for information security and IT governance to avoid such cases in the future. Organizations mature through experience, so consider this as an experience and learning from this incident will surely strengthen the company's processes.
Best Regards, Amod.