I think the answer to this will be “no,” but here goes.
Is there a way to provide a link that will go to a page or post showing the posts in a certain category? In other words, this would be the same result as clicking a category on the blog – except there would be a static link on a menu to get there, rather than steering the reader to go to the blog and then click the category?
If I understand you, you only need a custom menu, look under Apperance/Menus. Most themes support it.
That way you can make a category page which links from a menu tab, just like your home page and regular pages.