The difference between the US Operating Segment (USOS) and the Russian segment is one of self sufficiency.
The first module launched, Zarya, was enough to base a station on. Sure not a ton of living and working space but it had all the orbital control, solar, thermal radiators needed to be a space station.
After that, everything grew off that initial kernel with larger more competent versions of those subsystems being added. The obvious two are the huge solar arrays and the huge radiator fins.
Thus the Russian segment, has the solar, thermal, and orbital management it needs on its own at a small scale. But the US side relies on the bigger systems except it lacks the orbital management part. I.e There are no engines on the US side to reboost the station. But there are on the Russian side.
Since the USOS relies on the larger heat and power systems, that could not be provided on each node individually, it did not make sense to provide 100% self sufficiency on every module, since it is better to save mass/performance for the real purpose of the module.
As Nickloai notes in another answer, this is a difference between 1st, 2nd, 3rd, and 4th generation systems. ISS is a hybrid that started as a 4th gen station, and the USOS acts as add on modules to it.
Now if the US and international partners decided to go off on their own, they potentially could launch something like the ICM (Interim Control Module) and potentially move existing modules and components to a new base unit for a new station. They were not planned to do that from the outset, but if the new base unit provided the proper power, heat, and orbital control infrastructure they could unberth from one CBM port and berth to a second new one. If done while all attached to the station, they could likely even use the CanadArm to do the work. But the US, unlike the Russians, never seem to have planned for this.
Having said all that, your question is missing a subtle nuance. The Russian segment that would potentially separate and form a new Russian station post-ISS service life would be the newest modules, not yet delivered to the station (as of Mar 2015). They do not plan on taking Zarya or Zvezda with them. If they did, the US has the Interim Control Module (ICM), developed in case Zarya failed on launch to consider refurbishing to replace them.
The Pirs module docked to Zvezda's earth facing port is scheduled to be removed by a Progress departing the station, and the Nauka module is due to dock there. This is a similar module to Zarya. Then at the bottom of Nauka is the OM module, which is a 6 port docking sphere. It is ultimately the OM module that becomes the heart of the new Russian station, where modules come and go, but the OM remains.
There are two modules planned to dock to the OM that are not yet complete and may still change, as they have changed many times over the past decade.
There are some great images of the US and Russian segments in a question about whether the ISS will need more docking ports.