In 1998, Iceland approved GPS as a sole means of navigation for north atlantic flying...and I did so many times with (Icelandic registered) L1011 aircraft fitted with dual Honeywell HT9100 GPS units.
So, in answer to your question, yes, GPS is (has been) approved...since 1998, by some regulatory authorities.
NB.
I fully expect some will say...nonsense, however, these I expect are the same folks with the 'not invented here' mentality.