the_content() does not grab the PHP file content.php, it simply displays a Post’s content. Likewise, the_excerpt() grabs the excerpt of a post.
get_template_part simply finds a file within your theme named whatever you put in, with an optional suffix.
get_template_part( 'content' ); // content.php
get_template_part( 'content', 'my_page' ); // content-my_page.php
in order for make different displays for Gallery, Aside, etc. do this in single.php
get_template_part( 'content', get_post_format() );
It will point to content-gallery.php, content-aside.php, etc. with a fallback of content.php if the file specified is not defined.
Edit:
All you need to do here is restructure your content-{post-format}.php files. Inside those, use the_content(), and create excerpt-{post-format}.php files where you can then use the_excerpt().
Then your call in index.php would be get_template_part( 'excerpt', get_post_format() ), where inside your single.php it would be get_template_part( 'content', get_post_format() )