I've decided to answer my own question with my own speculation and digging.
Can the SIPRNet be physically distinct? Of course. Small networks can spring into existence without needing existing infrastructure all the time. Large networks are more of an issue. Telecomm companies around the world will lay a physically distinct cable for an outrageous fee. Something tells me that any significant military can also get the appropriate permits to lay their own cabling. Google is trying to purchase the rights to unused so-called "dark fibre" -- though that article is 6 years old, the thought remains the same.
But, for a network with global reach -- and as @TomLeek points out -- there are things that cabling cannot be used for. Satellite networks, mobile forces, and physically separated environments will need some form of through-the-air communications channel.
Because of the need for wireless communications and possibly because laying their own wires is cost-prohibitive (really?), not possible, provides no additional physical security over wireless, etc. a significant military power could share "public" infrastructure resources. It is likely such usage is only for low classified information transfer. In any event, any communications is going to be heavily encrypted using encryption algorithms and techniques that may not even be known (public/private key encryption was developed by signal intelligence researchers before DH and RSA were produced).
What I was really more interested in -- and the question was unfortunately worded -- was whether two separate networks -- SIPRNet and NIPRNet -- are technologically incompatible. All evidence points to "no, but". Any technological incompatibility would have to occur at the network edges (and definitely in layers, at that). The existence of products like a certified one-way cable hints at other physical mechanisms to ensure that non-secret and secret networks remain in a one-way relationship (information can be easily classified, but classified information is very difficult to declassify; eg. Bell-LaPadula: no read up, no write down). The existence of data-loss prevention software is a commercial realization of something that is likely already in effect in military organizations.
I think possibly more interesting is how three computers sitting on a desk remain separated. Are SIPR data cables physically locked into the data jack in the NIC? Do SIPR computers have no peripherals? I'd hazard a guess and say "yes" and "yes". Are the SIPR data jacks physically different? Maybe. I'd be interested to know the true answer to this which I don't think is classified information since there appears to be some reference to "SPIRNet RJ45" connectors in publically available military spec hardware products.
The following doc points to perhaps more of a policy-driven behaviour for attaching SIPRNet computers rather than a technological one. I also found that the following doc to be very interesting from a holistic view and this one to be interesting from a full design perspective of an installation.
All of these links and docs were found via Google. None are marked as classified which tells me that the separation mechanisms are probably working. ;-)