Using a shortcode to create a dropdown menu from wp_nav_menu list items

You can use wp_get_nav_menu_items() which retrieves all menu items of a navigation menu.
Your shortcode callback should be something like this .

  function print_menu_shortcode($atts, $content = null) {
       extract( shortcode_atts(
            array(
                'name' => null, 
                'class' => null
            ), 
            $atts
        ));

        // Assuming $name contains slug or  name of menue
        $menu_items = wp_get_nav_menu_items($name); 

        // Sample Output. Adjsut as per your exact requirements.

        // Sample Output variable.
        $menu_dropdown  = '';

        if ($menu_items){

            $menu_dropdown  .= '<select onChange="document.location.href=this.options[this.selectedIndex].value;">';

            foreach( $menu_items as $menu_item ) {

               $link = $menu_item->url;
               $title = $menu_item->title;
               $menu_dropdown  .= '<option value="' . $link .'">'. $title . '</option>' ;

            }

             $menu_dropdown  .= '</select>';

        } else {
           $menu_dropdown  = '<!-- no menu defined in location "'.$theme_location.'" -->';
        }

           return $menu_dropdown ;
}

More Examples on use of this function.