Theoretically possible I suppose, but it would assume that you knew with near-absolute certainty the wind vector and the compass deviations in the configuration that you were flying at. That and you could only use it if prepared to hold the beta angle long enough, in steady heading, for the GPS to stabilise. Can't see that working in real life - too many sources of error.
There is a tried and tested method of estimating Beta, by establishing steady heading sideslip, noting the compass heading, then swinging the aircraft back onto balanced flight and noting the new heading - the difference being roughly the sideslip angle. We tend to quote to the nearest 2° or so (giving about a 5° error bound), but realistically I'd not trust it to more than that. Good enough for testing the low speed aeroplanes I usually deal with that have can sideslip 30-35° degrees at full rudder deflection, but far too crude for anything fast that above Va probably won't slip more than 10° except as a very rough and ready guesstimate. Beyond that, my opinion would be that you need to bite the bullet and fit a yaw vane.
G