This is not automatic; it all depends. Simply being a director does not give a person the power to choose their successor when they leave office.
When a director resigns, dies, becomes incapacitated or unfit, or otherwise ceases to perform their duties, the method for filling the vacancy depends on the rules applicable to the company. Depending on the circumstances, the new director may be elected by the shareholders or appointed by the remaining directors, provided they have the required quorum and the law permits them to fill the vacancy.
An administrator who wishes to resign may therefore submit his resignation, but he cannot presume that he personally possesses the power to appoint his successor.
Regarding the election or appointment of the new replacement director, the required resolutions should be adopted and the relevant corporate records updated. Any required filings with government authorities should also be made.
In summary , a director does not automatically have the power to appoint their replacement. The rules applicable to the company must be reviewed to determine who can elect or appoint a new replacement director.