"My future mother in-law inherited a home in State X and wants to put it up for sale. Apparently there is a second partial owner who is in a nursing home. One of his relatives is contesting the sale. Does he have the rights to do so? There are no wills in place my mother in-law inherited the home by next of kin and owns the majority."
Next question is does the relative who is contesting the sale have the right to do so? If they are not the guardian or agent under a power of attorney for the co-owner in the nursing home they may have no legal right to any say.
If the relative contesting has no rights, and if the co-owner does own an interest in the house and is not competent (perhaps they are in the nursing home with Alzheimer's or another cognitive issue) then someone legally has to be appointed to represent his or her interest so you can sell (and that someone may not be the contesting relative).
Let's say you get through all of this and find that you want to sell and the person in the nursing home in fact owns an interest. You may need to hire a real estate litigator to bring a lawsuit to force them to sell. It might be a "partition" action. A claim brought asking to have the property sold and the proceeds divided. If there real issue is that they think the price is too low, you might be able to convince them with an appraisal and get them to agree to sell without all these complications. If they are objecting just to make it difficult for you and with no logical basis you might inquire of your attorney as to whether you can sue them for all the costs you've incurred in fighting with them. If they've done it just to be malicious a court might consider that.
There is another issue to consider. If the co-owner is in a nursing home, might the nursing home or the state have a claim on the interest the co-owner owns in the house for nursing home costs? While that may not be your issue, coming only out of the co-owner's share, it could complicate your sale.
Finally, we haven't spoken about taxes yet. It sounds like the probate process for the person who left the house to your future mother-in-law and the co-owner may not have been handled well. Your mother in law will have to determine what her tax basis is in the share of the house she owns so she can determine capital gains to report for income tax purposes when she sells (assuming she is not living in the house as a personal residence). That might be the date of death value of the house that the benefactor left her.