He keeps Gringo in conversation all the while having a gun pointed at him from under the table.
At any point Han could have shot Gringo.
He only shoots when the green guy tells him he is about to murder him for cash and steal his ship.
This is how Cold Blooded Han would go:
GRINGO : Going somewhere Solo?
HAN : Yeah
HAN SHOOTS GRINGO IN THE FACE, THROWS A COIN TO THE BAR KEEP.
HAN : Sorry about the mess