“Authenticate” Request When Submiting Theme Update

I posted yesterday about Featured Content Gallery not working and since then have found another problem that may be the root of the FCG issue. When I go to Update File in the Theme Editor, I’m taken to a page asking me to authenticate with a username and password. I’ve tried entering the WP login info and the FTP info, but am taken back to the original file in the Theme Editor with the changes unsaved.

This isn’t happening on any of my other sites, which are using the same theme and plugins. Thoughts?