Specific requirements still exist for DME for certain terminal procedures and although GPS is an approved alternative, it seems the FAA is not yet committed to eliminating them. As stated above this is due to potential GPS system failures and probably the recognition that there are still several aircraft that continue to rely on the DME system.
That being said, in today's environment, Part 91, mostly-VFR certificated aircraft operators could probably survive without their ground based DME equipment if they have a mostly reliable GPS source on-board.
• Install NextGen Distance Measuring Equipment (DME), in addition to Very High-Frequency Omni-directional Range (VOR) Minimum Operational Network (MON), to enhance infrastructure resiliency of Performance-Based Navigation (PBN) operations during Global Positioning System (GPS) outages.
The FAA Navigation Programs Strategy also addresses DME:
The overall objective for the NextGen DME Program is for DME/DME to provide a backup to enable aircraft to continue PBN operations during a GNSS disruption. Most commercial aircraft are equipped with GNSS and DME/DME avionics that can be used to continue flying RNAV procedures to their destinations and to an ILS or VOR approach (during a GNSS disruption). Under the Figure 11: Non Directional Beacon (NDB) NextGen DME Program, the HPDME network will be enhanced to fill coverage gaps and eliminate single-point failures....
(See, e.g., this PDF: https://www.gps.gov/cgsic/meetings/2019/lawrence.pdf)
You can use an IFR-approved GNSS (i.e., GPS-based "suitable RNAV system" as described in the AIM and other references) to substitute for DME.
For example, the January 30, 2020 udpate to the AIM includes a revision that clarifies that you can use GPS to substitute for marker beacons associated with ILS approaches (see AIM 1−1−9 and 1−1−10).
FAA is also charting stand-alone DMEs to support RNAV. See this item at my blog: https://bruceair.wordpress.com/2017/02/14/stand-alone-dmes-on-charts/.
Bruce Williams:You can use an IFR-approved GNSS (i.e., GPS-based "suitable RNAV system" as described in the AIM and other references) to substitute for DME.
...with one exception -- you cannot use GPS as a sub for DME for lateral guidance on the final segment of an instrument approach procedure, such as is done on the KMTN VOR/DME or TACAN Z RWY 15 and KWAL VOR/DME or TACAN RWY 10 approaches. Fortunately, those are, AFAIK, the only two such approaches in the USA using DME for lateral guidance on the final segment, so it's not a big deal in reality, but nice to know if you're on Jeopardy and have to take Instrument Approach Procedures for $500.