Hi flacoinohio. This is a tough subject. As a mother of a handicapped adult daughter (25), I had to declare her as incompetent and become her legal guardian. She has lived in a nursing/rest home for 4 years. Though she is coherent and able to make small decisions such as what to wear and eat, she still needs assistance in other things... especially when it comes to medical care. I decided to take over her guardianship during the first year of her living in the rest home as she was unable to make good decisions or care for herself without assistance.
As far as your uncle, you said that "he lives alone and is at risk of self harm and falling" and that "he is in need of constant care." If your uncle is unable to make mental decisions for himself, he can be declared incompetent and someone can become his legal guardian. However, he has many rights if he is still mentally capable of making major life decisions.
I don't know where you live, but many states and counties will provide medical assistance in the homes of the elderly and handicapped. Medicaid and Medicare will pay for this in many cases. Even local charities and churches may volunteer to take him grocery shopping or go shopping for him if his family is unwilling. There are even services (sometimes with his city or county) that will provide medical transportation. Are these options available for your uncle?