How do you define an orbit for property purposes? Please keep in mind that there are many, many orbits that intersect. That there are important orbits that may have many things in them. (Geostationary orbit is the most notable example.) That orbital mechanics are chaotic, so any object placed in orbit will eventually wind up in a different orbit. (One can, if you time it right, get from near Earth orbit to the Moon with almost no usage of power. IIRC this trip takes about 2 years.)
Next, once you've defined orbits in a reasonable way, you have the problem that most of the space junk out there will cross many orbits. For each owner of an orbit, it is easier to just avoid stuff crossing your orbit than it is to clean that stuff up. (Don't forget, we still have no way to clean things up.) Also any attempt to clean up junk that is crossing one orbit is fairly likely to move it into someone else's orbit. What are the lawsuit opportunities there?
Even if you propose a solution to these problems, you have the bigger problem of how to agree on regulations that have to cover multiple countries. Do you really think that China or India is in any mood to accept a division imposed by the USA, Europe and Russia about who may use what parts of space? And what about the smaller powers? According to Wikipedia there are 45 countries with space agencies. Many, admittedly, do not launch their own satellites. But getting that many countries to agree on a complex legal system involving space is going to be very, very tricky.
I agree that these are problems that need to be addressed at some point if we're going to continue our exploration of space. However I submit that there are a lot of issues to be resolved that free markets are poorly equipped to address. And even if you come up with a solution where free market forces have a role, you first need to create the framework that free market can exist in. And that process itself can't be solved by waving the magic free market wand.