And did he really accept them as "humans"?
If so why didn't he quit his job?
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Firstly, he did quit his job - that's the conclusion of all of the versions of the film. From the script:
Gaff: You've done a man's job, sir. I guess you're through, huh?
Moreover, he began the narrative having already quit the job, and had to be blackmailed into returning to it.
Deckard:I don't work here anymore. Give it to Holden, he's good.
Bryant: I did. He can breathe okay as long as nobody unplugs him. He's not good enough, not good as you. I need you, Deck. This is a bad one, the worst yet. I need the old Blade Runner, I need your magic.
Deckard: I was quit when I come in here, Bryant, I'm twice as quit now.
Bryant: Stop right where you are. You know the score pal. If you're not cop, you're little people.
Deckard: No choice, huh?
Bryant: No choice pal.
During his pursuit of the replicants, which involves him becoming familiar with the complexities of their daily emotional and practical lives, their histories, and their personalities, Deckard comes to the conclusion that replicants are at least as human as he is. This is particularly the case with Rachael, to whom he becomes sexually attracted, and who he eventually comes to love. In short, his empathy for the replicants changes his mind about whether it's okay for him to kill them.
That said, police officers are often called upon to use lethal force against people, so the issue of whether replicants are human may be less germane than the issue of whether it's ethical (or humane, if you prefer) to kill at all (especially as a job), or whether it's acceptable to kill people you empathise with.
SPOILERS BELOW. Blade Runner came out 32 years ago, but on the off chance you haven't seen it, look away now.
At the start of the film, Deckard is retired and reluctant to come back to work -- so he may already have been disillusioned with "retiring" replicants.
Throughout the film, there is a conflict between Deckard's desire to do his job, and his growing doubts about killing replicants. This is motivated in part by his increasing fondness (perhaps love) for the replicant Rachel. Also, Deckard may be starting to suspect that he himself is a replicant.
After Deckard kills Pris, he chases Roy through the derelict apartment building. He ends up clinging precariously to a rooftop, about to fall at any moment.
The replicant Roy Batty reaches down and saves Deckard's life.
Roy then dies. Importantly, Deckard does not kill him. Instead, he expires from "old age", because replicants have only a four year lifespan.
At the end of the film, Rachel is a fugitive. She is valuable property of the Tyrell Corporation, and it will want her back. Deckard has no intention of returning her. Instead they go on the run together, which (obviously) requires Deckard to quit his job.
Obviously, (4) is a key turning point. After that, Deckard doesn't want to kill any more replicants, so he will no longer be a Blade Runner.