I am guessing a bit but based on your description, you probably only have an index.php file running both of those pages. If you create a single.php file in the theme that (probably) matches what you had in the index.php before you edited it, you should have what you want.
WordPress will look in the theme for files of particular names to display content. Ultimately, if no specialized files are found, index.php will be used to display most everything. I think that is what happening in your case.
Another approach
Edit content.php and change the call to the_excerpt to
if (is_singular()) {
the_content(); // there may be parameters in your existing code
} else {
the_excerpt(); // there may be parameters in your existing code
}
Unless you have something like content-single.php in which case you will need to look in that file but it sounds like the same content.php is being used.