HTTP and HTTPS aren't interchangeable simply because one works. They are two different protocols, whilst both use HTTP to transfer data, HTTPS does this over a secure sockets layer.
Whilst the difference looks like a simple addition of the letter S, a lot more is going on. The server needs to be able to provide this particular service, many routers do not as the data isn't usually important enough to warrant the additional encryption (the most important data it'll be handling for most end users would be a Wi-Fi PSK, or an ADSL password).
As such, unless you're using a router that happens to have this somewhat trivial level of web management security, or it's an enterprise level router, you won't have this feature.
Just so you're aware, you will get a page not found message, as:
HTTP, by default, is listening on port 80.
HTTPS, by default, listens on port 443.
When sites use HTTPS, your browser will interpret this to mean https://192.168.X.X:443, as opposed to http://192.168.X.X:80.
As it's not listening on port 443, there's nothing to connect to, so it returns that no page is available.
As for what you can do about it, there are a few alternative firmwares, such as DD-WRT, Tomato or Open-WRT which may provide what you want.
The router's firmware needs to have that feature. Have you looked in the settings? – spherical_dog – 2016-01-09T16:31:29.707
I can't seem to find anything related to https in the settings. – conectionist – 2016-01-09T16:36:35.157
What actually happens when you try to access via HTTPS? Does it not resolve or do you simply get a certificate warning? Many routers use self-signed certs so the browser blows up with the necessary warnings, but you can go "advanced" and proceed over an HTTPS/encrypted connection if you want. – MrWhite – 2016-01-09T18:20:22.500
1Having said that, I also have a TP-LINK router and that is also not available over an HTTPS connection. – MrWhite – 2016-01-09T18:40:10.830
@w3dk It says "unable to connect" when I try over https – conectionist – 2016-01-10T10:09:49.677