An individual can make their own medical decisions including treatment and nursing home placement. If the person lacked the capacity to understand the nature of their illness and the consequences of refusing treatment (including placement), those decisions could be made by a designated health care proxy or under NY Family Health Care Decisions Act, by a family member. However, as a practical matter it would be difficult to place a person in a nursing home against their wishes. And you would have to deal with control over the persons finances in order to pay for the placement. A guardianship proceeding could be brought, and it would require a decision by the judge in order to place a person in a nursing home.
Even then as a practical matter, it is difficult to keep someone who is mobile in a placement they do not want.