Final answer:
Inspector Goole is the first to realize that Eric is the father of Eva Smith's baby in 'An Inspector Calls,' with Mrs. Birling inadvertently confirming this fact later on in the play.
Step-by-step explanation:
The first character to realize that Eric is the father of Eva Smith's baby in An Inspector Calls is Inspector Goole. As the sharp investigator of the events leading up to Eva Smith's death, Inspector Goole systematically reveals the connections between Eva and the Birling family, as well as Gerald Croft. It is during his interrogation that the truth comes to light.
Mrs. Birling, unwittingly, is the one who seals the connection. In Act Two, she declares that the father of the child should be held responsible, not knowing that Eric, her own son, is the person in question. It is during this dialogue that Inspector Goole, who had previously dropped hints, makes it known to the audience that he is aware of Eric's involvement. Later on, Eric enters and eventually confesses, and this confirms what Inspector Goole and the audience have surmised.