It would happen if there was enough demand for it, just like the first power plant, the first radio, the first car, the first steam engine, etc.

None of these items were developed in a vacuum; it's almost impossible to separate private and public contributions once you start talking about education and the public safety/infrastructure that created the right environment, but I don't think it's fair to imply SpaceX would never have happened without NASA or other public research. History, at least, has presented some counterexamples.