How can I write from the point of view of a character whose beliefs,
at best, I disagree with, and at worst, view as immoral and inhumane?
How can I give Day convictions in his beliefs when I myself have no
Congratulations. You've just realized that being a writer is harder than it looks.
This is one of the key questions you need to face if you want to become a good writer, and there is no cookbook answer.
It's easy to write stories in which the protagonist and all of the good guy supporting cast are essentially you, with perhaps some of your fantasies about yourself added just to spice things up. The problem is, eventually you wind up with a story infested with mini-me's and not much in the way of a believable story. Tossing in some of your friends will help, but only if you can do them justice.
In order to create believable bad guys, you basically have two choices:
1) Hang out with bad guys and get to know them. Well, OK, maybe just reading extensively about various types (especially if you read some apologists or fans), but that's just a different form of "hang out with".
2) Use your imagination.
And no, the second one is not (entirely) being snarky. I'm afraid my memory is a bit too rusty for proper attribution, but I recall reading a quote from an author who was being accused of sympathy for his villain's philosophy, and it went something like,
There is a term for writers who only write stuff they agree with. That
term is idiot.
The writer Joe Haldeman once wrote,
The conventional advice is, "Write what you know". This accounts for
the large number of bad novels written about middle-aged university
professors who are contemplating adultery.
On a slightly different tack, the science fiction writer Robert Heinlein had his greatest cultural impact with his 1961 novel, Stranger in a Strange Land which had a very considerable (semi-cult) following during the Countercultural Revolution of the '60s and '70s. Thing is, in an interview he stated that he basically wrote it as an exercise in embracing values he did not believe in, yet acting as a cultural agent provocateur.
So. Can you put aside your own prejudices and philosophy, and really and truly grant that others sincerely believe in something else? If so, can you (temporarily, one hopes) encompass those values? That, in effect, is what you need to do.
If you cannot do that, all is not lost, but it will mean that you'll need to avoid writing stories in which you (through your characters) must convincingly argue both sides of an issue.
EDIT - And, I would add, you should be aware that a book which attempts to discuss philosophy is one of the hardest types to do well, the other being comedy. Since a writer almost always favors one position over the other, it gets really difficult to do the other side justice, and the difference in the quality of the arguments shows up vividly. Plus, of course, the temptation to lecture usually becomes overwhelming, and knowing when to shut up tends to go by the wayside. ("But I have just one more point that NEEDS to be made!")
That's not to say that, even if you succumb to the temptations which occur, you will necessarily write an unsellable novel. Ayn Rand made a very respectable living that way.