Is it possible for Search Results not to include protected areas and content?

If the Search widget appears on a public page then a visitor can find the title and first few lines of protected posts. While they cannot access the full posts, even the title and few lines they see in the Search Results is more than they should. It contains confidential information and client names that we do not wish to be public. We have created public pages without the search widget but I have just discovered that when I access the site from my iPhone, the search box appears right at the top of every page, next to the 'Where to?' drop-down menu!