Force a browser to load the 'https' edition of a website, not the 'http'?

Yes, this can be done. The EFF has published a Firefox plugin named HTTPS Everywhere which contains a list of sites known to support https connections and will force the use of https when visiting those sites.

Even if you could find such a solution, you'd have the problem that some web servers will always answer https requests, but won't send you to the website's page. Most likely your browser will end up on some landing page of the web server. An application would not be able to distinguish a landing page from the page you actually want. I'd check the similar question for external programs... I'm not sure if you can accomplish this without an external program.

If you're using Firefox, check out the HTTPS Everywhere extension from the EFF.