Can a man keep his divorced wife on his health insurance if she can not work and can not get her own due to health reasons?

You need to call the insurance company that you are currently insured through and ask them to put her on a separate policy. And then you still pay for it. You may lose the family discount, but at least she may still be insured.

If it is a group policy, no. Coverage would be for spouses, and unmarried eligible dependents only. If he has a family (non group) policy that they are both covered under, they should be able to "split" the policy and each have their own separate policy, not subject to underwriting again. The combined premiums will go up, but the insurance company doesn't care who pays them, as long as it isn't an employer.