Can I create a page which shows only one category?

If you display a list of tags (via Tag Cloud Widget or wp_tag_cloud) or list of categories (via Category Widget or wp_list_categories) then clicking on one of those link will display posts in that tag or category. A Tag Template or Category Template can be added to control the look of that output.

See Template Hierarchy to see how WordPress determines what Template is used to display you posts in those conditions.

Thanks.

So can I create a static page whose content is populated with the category URL?

You said before, “Rather than create a static page”, but now ask “can I create a static page” ??

Please understand that you may not have to do anything if you just want to show a list of posts that belong to one category (or tag) because WordPress will handle that automatically for you. As an example, change to the WordPress Default Theme and then click on one of the links under the Category section of the sidebar–delivered to you will be a list or posts in that category.

So, not being sure what you are striving for…if you want a Page that shows in your nav bar, that then displays a list of posts in a particular category then read this FAQ: How to easily display links to both Pages and Categories in the blog navigation header?

Thanks – your last paragraph summarises what I want.

I’ll check out the FAQ.