There are several issues I regularly encounter with the WordPress Media library after a site has moved to SSL.
If you try to upload, and see the error message “HTTP error”, this points to a permissions error on the server. Sometimes when a site is switched to SSL, permissions have to be configured again. In most cases it’s a good idea to contact your hosting company to check your folder permissions. Please make sure the folder has the correct 644 permissions set, and that the website is looking at the correct uploads directory.
403 Access Denied
This is probably either hot link protection, or a .htaccess file in the uploads directory, or in the webroot, with hotlink rules in it. See also this discussion:
Media library from posts and pages shows empty, but general library shows all images
I’ve come across this issue once, this turned out to be the polylang plugin, where the option to translate media files caused this issue.