You might be able to, if you run your own Domain Name Server, however, nobody in the outside world will be able to access your new domain -- only the machines in your own network.
This is like saying: If I invent my own telephone, can I give myself a number of my choosing, splice it into the telephone line, and get people to call me?
No, because the routing of the telephone system (Internet top level domain system) does not recognize or attach to your invented system.