Cache issue.

Try changing

<link rel="stylesheet" href="index.css">

to

<link rel="stylesheet" href="index.css?v=<?php echo time(); ?>">

This will force the CSS to reload.

I have tried to create WordPress theme using your functions.php, style.css and index.php and for me, the stylesheet is loading correctly. Please make sure you have added your header.php and footer.php for the theme. Please study some tutorials for theme development on WordPress.You can look at this link https://codex.wordpress.org/Theme_Development for detailed information.

LPChip suggested me to use the debugger/inspector of Firefox and there, the head is greyed out. On further investigation, I noticed it said "css was not loaded because its mime type "text/html" is not "text/css"

That made me check something...

I got the solution:

I had an apache mod_rewrite activated. While the path was correct, it seems that this mod_rewrite or the dedicated php for the redirection somehow told the browser that the file was html not css.

From you comments on existing answers, it doesn't look like something we could find out based on the provided code. Here are a couple of things that could help you diagnose the problem though.

In your browser, open up developer tools. (For Firefox, get Firebug and hit F12, For Chrome or Opera, just hit F12 for the built in tools.)

Then you want to open up the network tab ('Net' in Firebug, 'Network' in Chrome)

Once you have that open, reload the page and look at the server requests. One of them will be for your CSS file. What you want to look for is the HTTP status. If that Status is anything starting with a 4, you've got trouble. Either it's a 403 which means you don't have the correct permissions to load the file, or you'll have a 404, as we all know, 'not found', which would indicate that your URL is faulty.

Another way to quickly find out whether the file is actually loading is to view the source of the page in the browser, and click the link to the CSS file. In Firefox, doing that makes the source viewer load the contents of the CSS file instead. If that doesn't work out, you also get an idea that it can't be accessed. It's worth noting that the first method will make it clearer what exactly the issue is though.

Most likely I assume you have a typo in your capitalisation of the file name or path. If it is hosted on a non-windows server, that almost certainly makes a difference.

I don't know if you try to add a css file in your theme or plugin. I will provide both examples. Use wp_enqueue_style in combination with get_theme_file_uri to enqueu a stylesheet in your theme directory.

For a theme, see example

function add_styles() {
    wp_enqueue_style( 'fontawesome-style', get_theme_file_uri( '/assets/css/all.css' ), array(), null );
}
add_action( 'wp_enqueue_scripts', 'add_styles' );

For a plugin, see example

function add_styles() {
    wp_enqueue_style( 'example-styles-plugin', plugins_url('/assets/css/admin.css', __FILE__), array(), null );
}
add_action( 'wp_enqueue_scripts', 'add_styles' );

For both, add an external URL:

function add_styles() {
    // Add Google Fonts
    wp_enqueue_style('google_fonts', 'https://fonts.googleapis.com/css?family=Poppins:300,500,700', array(), null );
}
add_action( 'wp_enqueue_scripts', 'add_styles' );

Try removing the leading slash. The leading slash indicates that the path starts at the website's root, which might not be the "Source Files" directory.

In other words, use css/reset.css instead of /css/reset.css.