Depending on what specific laws are used, he can claim parts of Mars. You can't claim the whole planet certainly. But if you had a robot colony that was mining a section of the planet making structures on it you could certainly claim that.

The big trick is enforcement, you need to be able to reasonably stop the use of your property for it to be your property. Normally on earth this just means that a government will use its framework of laws to do that, but US courts cannot enforce territorial claims outside of US sovereignty.

Right now what we are seeing is a big game of kick the can down the road. Until you get livign human beings onto a celestial body, anyone's guess is fair game. Perhaps you'll see some movement if asteroid mining becomes a thing in the near future, but I think we'll see a permenant outpost first due to the legal quagmire.