This can be done fairly straight-forwardly in SQL without a really complicated query, without subselects, and without requiring oracle:

Start at the times table, join to the schedule slots where the appt time is completely within the available time. Then try (LEFT) to join to that employee's appointments where there's any kind of time intersection. Now, in the WHERE clause, only accept rows where the LEFT JOIN failed (i.e. no conflicting appointments).

SELECT
  t.start,
  t.end
FROM appointment_times t
  JOIN scheduled_availability s ON
    AND t.start >= s.start
    AND t.end <= s.end
  LEFT JOIN appointments a ON
    a.EmployeeID = s.EmployeeID
    AND t.start <= a.end
    AND t.end >= a.start
WHERE a.EmployeeID IS NULL
;
[download]