The nursing home is threatening to discharge my dad if I don't pay for one-to-one care. I/he can't afford it. What can I do?

Call the VA and talk to a Patient Advocate.
Explain the situation.
It might be possible since the VA is paying for the Nursing Home that they could get him into the Care Center at a VA site.
If the Nursing Home discharges him where do they expect him to go?
You need to tell them, and tell the Patient Advocate at the VA that you can not take care of him and he can not care for himself. The Nursing Home can not just walk (roll) him out the door and leave him there.
When my Husband was in the hospital for a time they had a "sitter" but he was a med student and that was his "job" and he studied while "sitting" with patients.

OK, showing my dumbness here, but I thought that NH was the most care level? Where they provided all the care, as opposed to AL and such? If they can't handle your dad,, what is left, and why can't they do the job you are paying for? My GM was in a NH, and they did everything... Even when she tried to "escape" they did not ask for sitters and the like. What the heck are you supposed to do? Good luck with this!

I don't have the answer but it seems like this probably isn't a new situation to people in that business and I would guess there are people with experience and knowledge about options. I would sit down with the NH coordinator and ask them what the options are, neither your dad or you can private pay for one on one care, they know he is supported by VNA so what are your options? Also ask the people at VNA, they too must have some experience with veterans who have similar needs and know how others have gotten them met.