I don't think this fits in Nautilus. Nautilus is a file manager, your router is not a file.

You don't have to remember the IP address or even look for it, though, because you can just type http://gateway in your browser window, and it will work. I don't remember if it's systemd or NetworkManager which sets the "gateway" alias, but it has been around for few releases at this point. You can also use it in the command line, for example "ping gateway".

I hope this helps.