It's usually thought that the first stadium gig was the Beatles at the Shea Stadium.
But actually, stadium gigs go back much, much further than that.
The first stadium gig was almost certainly held during the ancient Games at Olympia -- a stadium gig if ever there was one -- at the world's first stadium -- though the musician is not known.
There were all sorts of side contests at the ancient Olympics. Some of them were definitely music contests. A few decades after the time of Nico & Diotima for example there was a trumpet blowing contest (and a documented winner). But there were certainly music contests at the Olympics long before.
The format was probably something like the battle of the bands events you see these days. It seems inevitable that the winner would have been invited to play at the closing ceremony.
The earliest stadium gig for which I can find the musician's name is the music contests held at the Pythian Games. The Pythian Games were played at the stadium above Delphi, beginning in 586BC.
The travel writer Pausanias had this to say:
According to the tradition the oldest contest, for which they first offered prizes, was singing a hymn for the god ... in the third year of the 48th olympiad, in which Glaukias of Kroton won (586 BC) ... they added a contest for singing accompanied by a flute and for playing the flute. As victors were proclaimed: Melanpous of Kephalai in the kithara-singing, Echembrotos of Arkadia in singing accompanied by a flute and Sakadas of Argos in playing the flute. This Sakadas won two more victories at the next two Pythian games.
Thus the ancient Greeks invented the stadium gig, even if we don't know the date or the first muso to get the gig.