Edward did go LOVER instead of FATHER. However, Edward's a seventeen-year-old boy from a very heteronormative time who I highly doubt would suspect what his sudden devotion of Carlisle's glorious face implies.
So, Edward said, "Ah, yes, this is my father"
And proceeds to be fucking weird about it for a hundred years.
The other caveat, is even if Edward did suspect this is not the norm of how people view directly related family members (or remembered his own feelings towards his biological father who died not too long ago and who Edward lived with for most of his life at that point) I doubt he'd be comfortable acting on it in any way or even admitting to it.
Edward's already this hideous, deformed, man-eating gargoyle and being what he'd view as a sexual deviant on top of that would not be something he'd want to admit to.
Edward Hates Life, the Universe, and Especially Carlisle
Edward is turned into a demon by this beautiful demon whom he doesn't really remember. The guy says Edward's family is dead, Edward's mother asked Carlisle to do this, and they now must leave.
And the man is causing weird rumblings in Edward's nonexistent stomach.
Clearly, this man is Satan.
Edward is deeply suspicious of his sudden feelings, his man-eating circumstances, his sudden misery and in an effort to loathe himself a little less chooses to loathe Carlisle.
Edward and Carlisle proceed to have a very tense relationship where Carlisle realizes that Edward loathes him entirely.
Sooner rather than later, probably long before Esme, Edward says, "Fuck you and fuck your starvation diet" and leaves to eat murderers and rapists.
Carlisle doesn't turn anyone else.
Edward Wants to Die of Mortification
Here, Edward's a little more open to the idea that he's in love with glorious, perfect, Carlisle Cullen. That face, that ass, that saintly disposition.
He tells himself that Carlisle must surely feel the same way. After all, after three centuries, Carlisle turned someone and it was Edward whom he had never really spoken to.
Surely, this is a mutual attraction!
Edward, being Edward, gaslights himself via his gift into believing Carlisle's totally into him. Carlisle watches Edward's face when Edward's not looking, he once thought that Edward's uncommonly attractive. Clearly, he turned Edward to be his lover and is too shy to make a move!
Edward waits for Carlisle to make a move. Then waits. Then continues waiting...
Then Carlisle turns Esme.
Edward tries to tell himself Carlisle just felt bad for this lady. She died a horrible death and he's just that nice. Then he realizes that Esme's in love with Carlisle.
And Carlisle still isn't making a move.
Edward now wonders if Carlisle's building himself a harem. He feels jilted! Betrayed!
He finally confronts Carlisle.
Turns out, Carlisle is in deep denial and claims he does not see Edward as anything more than a son. In fact, he says Edward is actually a little too young for him and had circumstances been different he would have not turned him at seventeen.
Carlisle's just not attracted to him.
This makes things... awkward.
Awkward and terrible as there's Esme, also in love with Carlisle, and certainly not giving up her feelings anytime soon. And she's older than him and clearly an adult, more mature, and a woman...
Edward goes off on his rapist eating journey, is miserable, but I doubt he can come back as the idea of Esme having married Carlisle in his absence, returning to that, haunts him.