No. Any company can use existing satellite launch companies to launch their constellation. You don't need to be a "rocket company" first. There are plenty that can do this on the cheap - ISRO for example.
My argument is about margin and replication costs not about if there are alternative routes towards replicating SpaceX's network, given enough money. Of course if you had infinite money you could burn up hundreds of rockets to get your network in place using existing providers. But my guess is that money is going to flow literally anywhere else if, for example, you will be spending 10x money as SpaceX just for table stakes of getting your satellites in orbit.
Nobody can do it as cheap as a used rocket. Since the they will be (relatively) mass producing their internet satellites, losing one on a highly reused rocket won't be much of a financial hit. SpaceX could test rockets to failure with their own sats for a tiny fraction of the cost anyone else would face.