How would I go about configuring Geoserver to only load from an https address? I have been combing the web and cannot find any solution.
I think that this question is not exactly related to geoserver.
You should configure your java application server (tomcat or whatever) to use https. Geoserver will inherit that configuration
If you are using tomcat you can try to follow this instructions. Or as stated in the comments, now geoserver doc also includes how to configure tomcat for ssl.