Sterling is clearly in the wrong here. Its immature and insecure of him to bring the issue up the next day against a player that came on in the 87th minute and was barely on the pitch. He is probably a little annoyed that social media mocked him a bit in the way Gomez picked him up like a little child.
However, Southgate shouldnt have dropped him. Its not that big a deal. Its a very Southgatey type to thing to do, being nice, formal, and politically correct but it wasnt necessary.
To put it in context Craig Bellamy once took a golf club to John Arne Riises's legs in a hotel in Spain. What did Rafa do? He played both players the next day. Bellamy scored and we beat Barcelona 2-1 at the Nou Camp! Bellamys goal celebration was a golf swing