My father lives in NC and until a medical incident lived alone. I live in Ohio and would like to move him here once his rehab is complete.

HE must be a resident.
Before considering doing this please compare the Medicaid programs of the two states, and please find out how long it will be before your father can qualify both physically and financially for Medicaid by Residency requirements.

Be certain that this is what you want to do, because once you take Dad into your home you have made it his home as well. I would caution you to please see an elder law attorney in your area to take your father at earliest convenience to get a care contract in place. He should have a part in paying shared living costs for housing (not rental as that is taxable) and for transportation, food and etc. That is fair and right and comes from his Social Security. Doing it any other way is gifting and will prevent his apply and qualifying for Medicaid in the future when he needs it should he need to go into care.

It is very very difficult to move seniors who are planning on depending on any governmental and state help from one state to another and Igloo, who is the best on this site for any of this stuff has made that point over and over. I would research all this VERY carefully before making a move. There is no hurry. Your father can go into care in his own state while all this is getting worked through.

You might want to think twice about taking him into your home. Caregiving a parent is a huge job, and their health only goes downhill. Goodbye to your life as you know it! Is that what you really want?

Also, the milder winters in NC are hugely more easy for elders to manage than winter in Ohio. He may hate it. Then what?

No, it follows the applicant.

One must qualify financially and medically for Medicaid. In most states it only cover LTC (so not AL or MC). LTC is assessed as medically necessary by his doctor, or in a discussion with the facility. Don't move him to any facility that does not accept Medicaid recipients. Ask first. In most states the financial app "look back" period can be 5 years. One has to have less than $3k in total assets (not including 1 house and 1 car) but this also varies by state.

From an online search:

"Yes, you can apply for Medicaid in Ohio right away after moving there, as long as you meet the eligibility requirements:
You are a US citizen or meet Medicaid citizenship requirements
You have a Social Security number
You are an Ohio resident
You meet financial requirements 
You can apply for Medicaid in Ohio online at Benefits.Ohio.Gov, by calling the Ohio Medicaid Consumer Hotline at (800) 324-8680, or in person at a Neighborhood Family Service Center. You can also visit your local library for help applying. 

To apply for Medicaid, you'll need to provide documents that prove your identity, immigration status, income, assets, child support, housing, and utilities, and disability. 

Medicaid eligibility requirements vary by state, and you can't receive Medicaid benefits in more than one state at a time. You'll need to close your Medicaid case in your previous state and reapply in Ohio. However, most states allow retroactive Medicaid coverage, which can cover up to three months of medical expenses before you apply."

It can be done but will not be simple or straightforward. You could use your OH address as his new address.

So he is currently a rehab patient in a SNF? A post hospitalization discharge to rehab path done, correct? & owns a home in NC? My answers based on this being the situation….. As a first step, pls speak with PT & OT doing his rehab. You want to try to clearly find out IF in their experience that someone like your Dad would be likely to be able to 1. return to his old living situation OR 2. will be “at need” for custodial care with its 24/7 oversight done in a NH or even Memory Care, OR 3. be ok in Assisted Living or an Independent Living apt with lots of amenities. Find that out as it’s really important because what his assessment is for care level “at need” will determine what programs he will qualify for….. whether in NC or OH.
Ask if Dad would be ambulatory enough for a car trip from NC to OH, like ask what his “transitioning” capabilities are, like if he needs an aide or a spotter to go to the bathroom, shower, get dressed, eat.

If he is in rehab as a post hospitalization patient, then rehab is being covered by his health insurance. Like Medicare & supplemental health insurance or Medicare Advantage plan. Facility is happy as health insurance pays way waaaaaaay more than Medicaid pays for someone there as a custodial care resident. Rehab will not go on indefinitely. Important that therapists speak with you abt how it’s going and when he is likely to time out in his progress. In theory 100 days but that rarely happens, usual 3-5 weeks. Rehab time is your window to make decisions as to where he goes after rehab ends.

Medicaid is a huge # of different programs in each State. If he is low income, he may be already on NC Medicaid as health insurance. He is a “dual” so on Medicare & Medicaid. For duals, moving States and getting onto Medicaid for health insurance in the new State is pretty easy as it’s established he’s low or lower income.
HOWEVER
its the LTC Medicaid program that pays for facility costs AND a different Medicaid program. LTC has both medical & financial “at need” requirements which varies by State as each State administers Medicaid under overall Federal guidelines. You have to look at requirements for LTC Medicaid OH to see if he can qualify both medically & financially.

HOUSE: if he has a home in NC, will be an issue as non-exempt asset for OH and have to be placed For Sale with $ used as his spend down. This isn’t uncommon, what tends to happen is NC house placed on MLS with a Realtor and he is “LTC Medicaid Pending” till the sale. One big issue POA/family are gobsmacked at is that LTC Medicaid requires Share of Cost from day 1 of almost all his income to the NH. Less a small Personal Needs Allowance (avg $50/$75). So he will have zero $ for house in his name in NC. So on you to pay for whatever needed to be market ready and any costs occurring till sold. And be ok to cover costs without expectation that you will be easily reimbursed as that appears to be gifting.

Medical at need: If medically he is assessed ok for AL or MC and NC & OH does not do waivers from LTC Medicaid program to pay for these, then it’s private pay. LTC Medicaid is dedicated funding under Federal law for SNF custodial care costs. AL and MC are not unless a State does waivers. Facilities do not have to participate in LTC Medicaid, and some that do have a limited # of beds, so it’s on you to find out how feasible placement will be for him if he has to be in a facilty.

His legal and banking will need to be changed to OH. Hopefully he still appears competent and cognitive enough to do what’s needed. New Banking will mean he requests a change in his SSA & he should also file SSA paperwork to have you named to be his representative payee if he hasn’t done this already. All things that can be done but not simple or without co$t$.