No you may not legally do this. Federal law requires you to pay every employee for all hours worked. Regarding state employment laws, even if you do not live in an employment-at-will state, no law requires employees to give you notice when they want to quit.

The most you can do to reduce the possibility of this occurring is to have promulgated policies in place that specify that accrued, unused vacation/sick time that might otherwise be paid upon termination may be withheld if proper working notice is not given by the departing employee.