I'd like for my website to not have a domain name, and only use the IP address. However, when I go to register a website, the service provider demands a domain name. How would I go about registering an IP address without a corresponding domain name?
You don't need a domain name to access a website. You just need a web server with a static IP and every time you want to access that server you should remember that number. That is what the domain name mainly does, translating those sequences of numbers into a readable, easy to memorize name.
To put it easy, you don't register an IP address, you use the IP address that a web server has.
So, don't register a domain name, just use some type of web hosting service and note down its IP address.
I would like to point out also that in case of a shared server or if using subdomains, you would need to have a domain to help the server locate your desired website, the above answer applies to a dedicated server or if the IP resolves directly to that website, that is, it has just one website.