That is not a viable solution for me because I use the image elsewhere so I don't want to delete it from the media library.
However, with a little trial and error and browsing in phpMy Admin, I figured it out. Here's what I did for anybody else in the situation.
I went into the wordpress database and to the table _postmeta and looked in the meta_key for content with the word "header" and discovered two entries that had "_wp_attachment_is_custom_header" as the meta_key. As I had two images currently uploaded as "headers" this seemed promising. I looked at the post_id in the two entries and the post_id corresponded to the post_id of the two header images. Even more promising! I went back into wordpress, uploaded another image as a header, and voila! A third entry in _postmeta with "_wp_attachment_is_custom_header" appeared. I had it!
So, I simply deleted the table entry for the image I no longer wanted WordPress to consider a header, refreshed the header configuration page, and the image was no longer displaying as an uploaded image. I checked my media library and the image was still there (yippee!) but it no longer had "Header Image" noted next to the file name.
Not for the faint of heart, but worth it so I didn't have to delete and re-upload the image and fix links to the old image across the site.