Resignation is the voluntary termination of the contract of employment by the employee and it takes effect only when it is accepted by the employer. If it is covered by any specific term relating to notice period in the contract, employer would normally accept it subject to the fulfillment of the notice condition as mentioned in the contract only or at times may waive it. It is a matter of the employer's exclusive discretion whether a request is made in this behalf by the employee or not.
In your case, the resignation submitted on 09-06-2018 was accepted and he was actually relieved on 16, June 2018 in spite of the employee not serving the one month notice period or not paying salary in lieu thereof. It would certainly mean implied waiver of notice condition by the employer. Therefore, the ex-employee's demand for proportionate salary for the days he worked in the month is correct and justifiable as well.
6th July 2018 From India, Salem
Company having accepted the resignation cannot adjust days worked against notice period.
Question of notice period does not arise here as company is the authority to waive it off by express words or by action(here action of accepting resignation)
Good employers do not quibble over such small matters of with holding 18 days pay.
Pay him and let ex employee be happy.
6th July 2018 From India, Pune