That may well be true, but FAA AC 90-100A of March 2007 has "harmonized" RNAV and PRNAV requirements, standards, and procedures with ICAO standards.
AFAIK, ALL RNAV SIDs and STARs in the US require RNP-1 or RNP-2, and that the aircraft systems and/or operators be certified for them. All current RNAV SIDs that I can find at LAS are RNAV1.
While GPS may not be required for RNP-1 certification and operation, in the US a pilot still cannot legally fly an RNP-1 / PRNAV procedure unless the aircraft is certified and the equipment is operating in a mode that is still certified for it. From par. 10.a.(4) and 10.b of AC 90-100A:
(4) If not equipped with GPS/GNSS, aircraft must be capable of navigation system updating using DME/DME/IRU for RNAV 2 or RNAV 1 routes, as well as RNAV 1 Departure Procedures (DPs) and Standard Terminal Arrivals (STARs). As stated in paragraph 8.b.(1), if the navigation system does not automatically alert the flight crew of a loss of GPS/GNSS, the operator must develop procedures to verify correct GPS/GNSS operation.
b. General Operating Procedures. Operators and pilots should not request or file U.S. RNAV routes or procedures unless satisfying the criteria in this AC. If an aircraft not meeting these criteria receives a clearance from ATC to conduct an RNAV procedure, the pilot must advise ATC that he/she is unable to accept the clearance and request alternate instructions.
If you read further, you will find that for DME/DME only updating, manual runway position updates on takeoff are permitted, but auto-tuning of DMEs for updating is required. So, again, you have to look into your Flight Handbook, DDPG, and OpSpecs (or equivalent) to ensure you retain the required certification for a degraded system.
While "737incognito" is correct that many European SIDs and STARs are BRNAV only, that is because they are overlay procedures that can be flown with radio NavAids in the terminal area and require BRNAV for transitions and enroute.