Is it legal to leave my dad alone?

I would think this depends on his level of "incapacity"
Can he make rational, competent decisions in most cases?
Is there a potential that he would hurt himself if he were left alone?

This is like asking if you can leave your child home alone.
a 4 year old, not under any circumstances.
a 10 year old, possibly depending on how long you will be gone and how responsible they are. Responsible and left alone for an hour or two, not a problem but not an entire day.
You know the level of direction your child needs, same with an adult with dementia.

You read many posts here where an elderly parent is living alone. While not a great idea but it does happen.
Use your best judgement. If something in your gut says it is a bad idea, listen to your gut.
Try an hour of shopping and see what happens.
Then try a lunch.

No, unless you are guilty of neglect, which this certainly doesn't sound like at all. That being said, there is always a first time an elder decides to walk out the front door and then not remember why they did or where they live. Or turn on a stove burner, or think they didn't take their meds that day, etc. Just because he's urging you to do it doesn't mean you should. As our elders decline, so do their abilities of reason and logic. Only you are the best judge of whether it is worth the risk.

I don't know what the 'legalities' are of leaving an incapacitated elder alone, but I would hire caregivers to come into your home for 4 hours a day a few days a week to start. That will acclimate him to having others around to help him b/c you simply cannot BE there 24/7, it's unrealistic and harmful for your own state of mind to do so. An elder with dementia can get into trouble alone at home in a New York minute, so while it may turn out okay to leave him alone while you go for a walk, you probably wouldn't feel comfy doing so ANYWAY, so why do it? Hiring in home help with dad footing the bill makes the most sense b/c sooner or later, it'll have to happen so you can go out for dinner and a movie or have time alone. As dad's dementia worsens, you're going to need help with his care and management, be assured of that. Having him get to know caregivers and be familiar with them is a very helpful tool to have in your toolbox moving forward.
Know what you're in for and educate yourself on the subject here:

Understanding the Dementia Experience, by Jennifer Ghent-Fuller 
https://www.smashwords.com/books/view/210580

That 33 page booklet is a good start, and The 36 Hour Day is another good reference book to read and keep handy.

I would check the elder laws in your state about what constitutes neglect, or contact an Elder Care attorney for clarity on that subject, just to be sure. The elder will always 'tell you' he's fine and doesn't need any help at all! That's the nature of dementia, and denial that anything is wrong to begin with. They often think they're capable of WAY more than they actually ARE; that's where the trouble starts, too.

Wishing you the best of luck with everything you've taken on.

Not sure if its illegal, but I would not do it. Dementia is very unpredictable. You never know what they will do. They are not aware of their limitations.