Is a WAAS GPS required to fly in RNP airspace (e.g., RNAV (GPS) approaches)?
Or, can you be approved to fly RNAV (GPS) approaches with, say a military EGI (embedded GPS/INS)?
This is a random RNAV (GPS) approach. The answer is it depends on the minima, I highlighted above two examples.
The LP (WAAS / SBAS) approach requirements can be found in the FAA AC 90-107.
To fly the LP approach, the GPS unit on your plane (among other things) needs to be LP certified. For example, a Garmin GNS 480 unit is LNAV/VNAV approach approved, and only LP approved if you have software 2.3 or later installed.
If you want the regulatory requirements, then check the AC linked above and all the referenced technical documents (such as TSO-C146). If you want to check the capability of your GPS unit, then refer back to the manufacturer.