My father was denied extension of insurance coverage for rehab post hospitalization due to insufficient progress. What can we do?

Another vote for you to appeal. Include any documentation from the rehab facility as supporting evidence that he didn't/couldn't get the services.

https://www.patientadvocate.org/explore-our-resources/insurance-denials-appeals/things-to-include-in-your-appeal-letter

From your profile: "I am caring for my father Charles, who is 75 years old, living at home with cancer, hearing loss, heart disease, mobility problems, and stroke.?

Does he live with you?

Just remember that it's not a choice between rehab and home. It's a choice between rehab (covered my Medicare) and becoming a Long Term Care resident paid for with his funds, or by applying for Medicaid if he can't afford to private pay.

If there is a spouse involved who will be affected by a Medicaid application, get a Medicaid planner or eldercare lawyer involved immediately so that assets and income can be divided so as not to impoverish her.

Appeal, and you can try the unsafe discharge route. One week in isolation because of Covid should be taken into consideration.

File an appeal. The rehab should be able to help you with this. Otherwise the back of his EOB should give you instructions. You have to appeal within 3 days of denial of coverage.

When you file an appeal, it keeps them from sending him home, so appeal the appeal answer, if it isn't yes and again, if needed. I think it takes several appeals to get insurance companies to change their minds about coverage.

Is he making progress? If not, why not?

I would talk with your FIL and make his choices clear. If he cannot be mobile then he may end up in long term care. Only he can work hard enough to keep his choices open.