I recently saved a very young baby coyote from death. A hawk or owl got him, and he laid in someone's yard for two days. We were working on a house next door and the lady came out asking us to get it out of the yard. Not long after that the poor little guy got maggots in his back. I sat tediously at my big kitchen sink all night pulling them out and I was wondering if taking care of this little guy and nursing him back to health will tame him?
This is a spectacularily bad idea and would probably lead to the death of the coyote anyway.
The best thing you could do is contact a wildlife rescue center as quickly as possible and let them take care of the pup in way that allows it to be released back into the wild. The longer you keep it, the more it gets used to humans. It would lose its natural shyness and cause trouble by rummaging through trash cans and killing peoples pets.
This person (who is certified to keep coyotes) explains in her video (I resorted some quotes to keep it short and in context):
[...] it's a 24/7 operation and they're never gonna be exactly like a dog. Even if you do have 24 hours a day to spend with them and have them inside, those folks usually have to cage them at night or, you know, when they're not in actual training, because they're still wild. They're still wild animals. These guys, sadly, were found as babies and exposed to people and a little too tame to be released back into the wild. So, that's why they are in a facility like this.
[Keeping them] contained is a big issue. So we have fencing [she shows horizontal fencing protruding inwards from the top of the vertical fence]: on top we have 4 feet, [on the ground] we have 4 feet - so they can't dig out - of fencing layed on the ground, covered by stones, because they are diggers.
You don't really want them around children. [...] You know, they'll go after other animals, too. [...] This man would eat all my cats over food, and over just a prey drive. [...] Sadly his life is in a cage. [...] This is as good as it gets for some coyotes and he's really lucky because wild coyotes that end up in human hands end up euthanized, cause not a lot of people are able to provide them, you know, with a life long space and semi-captivity. Zoos don't want them.
It's a better idea to go to a shelter and save a dog that really needs a home than try to take a wild animal and force it to be somewhat domesticated.
I would say while it is theoretically possible, in practice it would almost surely prove to be a bad idea.
Firstly, depending on your location, it could turn out that your local law prohibits keeping wild animals as pets at all.
Secondly, even if it was the case that private possession of such animals was legal, coyote pup is still a wild and unpredictable creature. For this reason, even if you decided to put an enormous amount of effort in tedious attempts to tame a wild coyote, it would probably turn out to be unreliable at best and your general experience would be inferior to experience associated with having for example a dog or a cat. In other words, even despite your best efforts and intentions it just wouldn't be any good as a pet and wouldn't have the desirable qualities we love our dogs and cats for, but instead it would have a wide variety of extremely undesirable and hazardous ones. Please mind the fact that for comparison it is estimated that it took about a few tens of thousand years to domesticate dogs from their wild ancestors and this long process, involving genetic trait selection, just can't in any way scale down to a single specimen.
Also, coyotes are carnivorous predators and their hardwired killing instinct would make them extremely dangerous in domesticated settings, because it could and would unpredictably switch on at any time. I've seen photos documenting injuries that wild coyotes are able to cause to people, cats, dogs, deers, etc. and in case of animals they sometimes were fatal, it wasn't any nice, trust me.
What is more, as far as I know it would be much more difficult to find a coyote-specialized veterinarian. And said veterinarian would definitely have to be involved in the taming scenario because of the wide plethora of pathogens and parasites a wild pup like this could have and potentially transmit to other animals and people.
I want to thank you really much for your good will and the tedious effort you've put into saving the life of this poor soul, removing the maggots and taking care of it. But after you're done with the immediate-response emergency healthcare for this pup, the best thing both for you and for the coyote would be involving a specialized wildlife rescue organization and letting them take further care of it.