Can I change all my http:// links to just //?

I tested it thoroughly before publishing. Of all the browsers available to test against on Browsershots, I could only find one that did not handle the protocol relative URL correctly: an obscure *nix browser called Dillo.

There are two drawbacks I've received feedback about:

Protocol-less URLs may not work as expected when you "open" a local file in your browser, because the page's base protocol will be file:///. Especially when you're using the protocol-less URL for an external resource like a CDN-hosted asset. Using a local web server like Apache or IIS to test against http://localhost addresses works fine though. Apparently there's at least one iPhone feed reader app that does not handle the protocol-less URLs correctly. I'm not aware of which one has the problem or how popular it is. For hosting a JavaScript file, that's not a big problem since RSS readers typically ignore JavaScript content anyway. However, it could be an issue if you're using these URLs for media like images inside content that needs to be syndicated via RSS (though, this single reader app on a single platform probably accounts for a very marginal number of readers).

The question of whether one could change all their links to be protocol-relative may be moot, considering the question of whether one should do so. According to Paul Irish:

2014.12.17: Now that SSL is encouraged for everyone and doesn’t have performance concerns, this technique is now an anti-pattern. If the asset you need is available on SSL, then always use the https:// asset.

If you use protocol-less URLs to load stylesheets, IE 7 & 8 will download them twice: http://www.stevesouders.com/blog/2010/02/10/5a-missing-schema-double-download/

So, this is to be avoided for CSS if you like good performance.

Yes, network-path references were already specified in RFC 1808 and should work with all browsers.

Is this completely cross-browser compatible? Are there any other caveats?

Just to throw this in the mix, if you are developing on a local server, it might not work. You need to specify a scheme, otherwise the browser may assume that src="//cdn.example.com/js_file.js" is src="file://cdn.example.com/js_file.js", which will break since you're not hosting this resource locally.

Microsoft Internet Explorer seem to be particularly sensitive to this, see this question: Not able to load jQuery in Internet Explorer on localhost (WAMP)

You would probably always try to find a solution that works on all your environments with the least amount of modifications needed.

The solution used by HTML5Boilerplate is to have a fallback when the resource is not loaded correctly, but that only works if you incorporate a check:

<script src="//ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"></script>
<!-- If jQuery is not defined, something went wrong and we'll load the local file -->
<script>window.jQuery || document.write('<script src="js/vendor/jquery-1.10.2.min.js"><\/script>')</script>

I posted this answer here as well.

UPDATE: HTML5Boilerplate now uses <script src="https://ajax.googleapis.com/ajax/libs/jquery/1.10.2/jquery.min.js"> after deciding to deprecate protocol relative URLs, see here.

I have not had these issues when using ://domain.com - but you do need to add the colon at the beginning. Yoast had a good write up about this a while back. But it's lost in his pile of blog posts.

If you would like to make sure all requests are upgraded to secure protocol then there is simple option to use Content Security Policy header upgrade-insecure-requests

Content-Security-Policy: upgrade-insecure-requests;

https://developer.mozilla.org/en-US/docs/Web/HTTP/Headers/Content-Security-Policy/upgrade-insecure-requests