No, shareholders elect the Chairman and the Board. Company founders often serve as CEO and Chairman in earlier stages of company growth and may remain the Chairman but give up the CEO position as the company grows bigger and requires someone more suitable for the CEO position (not in all cases obviously).
Apple's Board and shareholders are likely content with Levinson as Chairman and see no need to elect Cook to the position.
Not every CEO is the chairman. They perform different functions and while there are some who do both, its up to the shareholders to elect.
The CEO position is the top management position of a corporation where as the Chairman is an executive position to oversee the interests of the shareholders. That is they report to the shareholders and the CEO reports to the chairman.