I was really hoping someone would jump on this. I think this is going to take a bit of unwinding so here goes...
IEC 17025 is just a standard for providing test services (among other things). Many companies that are not NRTLs are certified to 17025.
An NRTL is a test lab recognized by OSHA and allows the lab to give authorization to apply a safety mark (e.g., UL, ETL, etc., for compliance to 60601-1). I don't think a test lab can be an NRTL without certification to 17025.
My understanding is that, in the US, hospitals won't allow devices in unless they have a safety mark. I've never tried it, but it *may* be possible to submit data for FDA clearance (presuming such data is required) demonstrating compliance to 60601-1 without going through an NRTL. (And I know you can do the testing outside of an NRTL, submit the data to an NRTL, and if all is good, get approval to apply a safety mark).
I *think* this is how it is; hopefully if someone knows otherwise, they'll weigh in.