It is. And increasingly so in fact, as it allows more airports that don't have the resources to install and maintain ground-based equipment to have previously unavailable approaches. https://www.flyingmag.com/everything-you-need-to-know-about-...
However, it doesn't fall into exactly the same class as ILS approaches.
WAAS (GPS plus a correction signal transmitted by separate geostationary satellites) improves the resolution to CAT I standards - an "LPV" approach. However, it's still not sufficient for CAT II/III approaches.
A CAT III ILS approach requires very specialized equipment and training because you basically fly it all the way down to the runway with no requirement for visibility until moments before touchdown.
However, GPS is a lot better when it is not being obstructed by buildings around you. Planes have the benefit of unobstructed paths to GPS satellites and would get a better quality and accuracy then you might get with a phone in a city. Also, using good antennas helps. So, the altitude accuracy would be fine mostly.
I.e. the transmitter on the ground must be able to detect that something is wrong and immediately cut out the signal / report it as inop to the receiver on the plane. When you're seconds from hitting the ground, that is a requirement.
Although GPS based approaches have some aspects of this, I believe that they just don't meet the exact requirements as written currently, although they are close.
ICAO recognizes SBAS Cat I as a precision approach, and there are dozens of them in Europe (France, Austria at least).
In good conditions the approaches might be indistinguishable, especially given that GPS approach once on final will be usually a copy of GPS one, but the safety of GPS LPV is considered much lower, for a good reason.