If a SID chart says "GPS... required", can the plane use a different GNSS?
The FAA's standards require the use of GPS, i.e. the American satellites. This is covered in AC 20-138D, Airworthiness Approval of Positioning and Navigation Systems. It defines GPS:

k. Global Positioning System (GPS). GPS is a U.S. satellite-based radio navigation system that provides a global positioning service.

It also says that GLONASS and other non-US systems are not supported:

GLONASS is not yet approved for IFR operations in the U.S. National
Airspace System
[...]
No FAA TSO exists for GLONASS, GPS/GLONASS, multi-constellation or
dual-frequency avionics. Therefore, adding capability for any other
constellation or dual-frequency capability must be accomplished as a
non-TSO function until appropriate TSOs are available.

Apparently there is some kind of working group looking into using non-US GNSS but it's not possible yet. See Appendix 8 of the doc for full information.