Can the state decide to put our grandmother in a home without giving the family any other options?

You do not understand what the doctor said. He is not saying that he can take your grandmother out of her current home and place her in a FT care facility. Only that your family cannot put her in a FT care facility WITHOUT him giving his approval. FT care facility places are very very hard to find and they are competitive - so to ensure that only the most needed people get the places and not those whose families "want to get rid of granny" the doctor has to write a referal stating that the persons needs are soo great they cannot be looked after by family anymore (e.g. dementia).