What if someone refuses to go to a nursing home?

Your profile says this is your WIFE:

"I am caring for my wife Alice, who is 81 years old, living at home with age-related decline, alzheimer's / dementia, depression, diabetes, hearing loss, incontinence, mobility problems, sleep disorder, urinary tract infection, and vision problems."

Does she have a medical diagnosis of dementia from her physician? If so, this is helpful. Are you her PoA? If so and she has a diagnosis of dementia then you can legally act to transition her to a facility but will need to work with that facility to get her there without a struggle.

If she doesn't have an actual diagnosis and you are not her PoA, then if she threatens to physically harm you (or actually does it) then call 911. They will most likely remove her from your premises to a hospital. At that point you can tell the hospital she is an "unsafe discharge" and that you are not able to care for her. If you are not her PoA then either you or the county will need to acquire guardianship in order to manage her medical and financial affairs.

Call Adult Protective Services and report her as a medically vulnerable senior with no help

People retain the right to make their own choices - even if those choices are bad. Two possibilities. She could assign guardianship to someone willingly with a power of attorney. Otherwise only a judge can transfer guardianship of her to another person.

Without guardianship, no one can legally force her to do anything at all. Elder abuse laws protect seniors from abuse but also prevent loved ones from doing things the senior doesn’t want but are obviously in the senior’s best interest.

I had to lie to my dad to make him move out of my apartment and into a care facility. It was for his own good so I don’t feel at all bad about it.

Call APS for wellness check. They can give you some guidance. Let them know you are walking away from helping at home. Truth is that Beatty here is right in that "there will be no solutions if you are all the solutions". Let APS know that you suspect she is unsafe and need guidance about how to get her assessed. If they agree an ambulance should be called they will get you referred to social services who can get you appointed as temporary guardian if need be. HOWEVER, do know that if she fights guardianship in court she may win. The state is loathe to take a citizen's rights from her.
There really is no good answer here. If the state is made guardian they will place her, but at that point you will have no say about where she is placed or what fiduciary manages her assets or how they are managed.
Are you alone in being responsible or are there other siblings nearby.
Wishing you luck, hoping you will update. Sometimes there are no answers until an elder does eventually and INEVITABLY come to grief from being alone and having an accident. We have even seen cases on Forum where a child tried with all her might to get help and her mom died in the home without help. It is very sad and tough. Again, I am so sorry.