Imagining his smoldering eyes and dashing looks, it's easy to forget that Edward is a 100+-year-old man. With age comes wisdom, but also a certain reluctance to change your views about the world and yourself.
Although there's no doubt in Edward's mind about how much he loves Bella, he also continues to think of their relationship as wrong, because he poses a danger to her. A part of him still sees himself as a monster with no chance at redemption, and he therefore believes that he has done her more harm than good. Although Bella is dead-set on her decision to be transformed into a vampire by Edward, he struggles with his role in that decision. He hates taking away her ability to have a human life and children, saying, "I want to give you things, not take them away from you" (2.57). On the other hand, he doesn't want to make love to her until she's a vampire, because he doesn't trust himself to control his killer instincts in the heat of the moment.
After he makes love to Bella for the first time, Edward feels absolutely terrible about the bruises he inflicted on her, and vows to not make love to her again until she's been changed. "I will never hurt you again" (5.220). Despite the continued self-torture, he does admit to Bella that making love to her was the "best night of his existence" (5.170). So Bella keeps trying. After putting on sexy lingerie doesn't do the trick, she begs, and it works like a charm. This time, Edward proudly manages to not harm Bella and warms up to the whole lovemaking thing. Just in time for Bella to tell him she's pregnant.
To say that Edward is shocked by Bella's announcement would be an understatement. He goes into full-scale paralysis: "A vampire who was still frozen on the floor with no sign of ever moving again" (7.71). When he finally does comes back to his senses, things seem clear to him: he's a monster and, thanks to him, there's a monster inside of Bella that is going to kill her. He promises her to get "that thing" out of her.
Edward's self-loathing and pain over what he has done to Bella (despite her insistence to the contrary), and his helplessness to fix his mistake, causes him to teeter on the edge of madness. Jacob describes him as a "burning man" when he first sees him at the Cullen house:
His black eyes burned in their sockets, out of focus… His mouth opened like he was going to scream, but nothing came out. (9.137)
In Bella's condition, Edward sees all his worst fears come true. As Meyer explains, "He's kind of a pessimist… always waiting for like the worst thing to happen." (source)
But it doesn't happen. Instead, Edward's whole perspective on his life with Bella and himself changes when he starts to hear the baby's thoughts and learns that it loves Bella. "He couldn't hate what loved Bella," Jacob realizes when he watches Edward finally falling in love with the baby in his wife's womb. Edward acknowledges that the baby is part of Bella as much as it is part of him. Making a personal connection with the baby helps Edward develop faith in their future and faith in himself as a good person and a good father.
His faith drives him to keep pumping Bella's heart after Jacob declares her dead, refusing to let her go. Or maybe it's just his natural role to bring her from the human world into the vampire world. After Bella's transformation, Edward also experiences his relationship with Bella as stronger and closer. He's immensely proud of her and of his daughter. Bella's ability to open her mind to him seems to be the last missing piece in the puzzle of their relationship and concludes his story.
For more info on Edward, check out his "Character Analyses" in the other books of the Twilight series on Shmoop.