And apparently it depends on aircraft type. I posted above the relevant excerpt from a EU-OPS approved 737 FCOM which states quite clearly "raw data monitoring is not required".
Of course the approach is predicated on the relevant nav aid, but that doesn't necessarily mean you have to check it if there are other means of compliance which assure you stay within the limits of the relevant approach.
In terms of RNP monitoring boeing makes it very easy, it directly displays your ANP and RNP as navigation performance scales on the PFD and on the legs page on the CDU. No need to care about GPS or RAIM, as long as your ANP is within limits everything is fine.