I've successfully implemented no password login via RSA key exchange to SSH on Linux many times. What I'm hoping to do is achieve a similar authentication for my HTTPS RESTful web services, but I don't know where to start, nor how to phrase the question in searches to get started in the right direction.
You're probably looking for SSL Client Certificates. Using client certs, your client can authenticate itself without interaction, with your API.
That said, this technology is not in widespread use, and as such, documentation can be lacking at times, and many HTTP client libraries won't even support it.