My mother was in rehab from falling. She can be cared for at home, but my sister doesn’t want to care for her so she put her in a nursing home. What can I do if my mom wants to come home now?

In the end it's a decision that you have to make as a family. Was she living with your sister? If so, your sister has some say in the matter, as she may not be able to care for your mother in this condition. Where would your mother get the best care? If she would be neglected or in danger if she left the nursing home, she is probably better off there. Only you and your family can decide.
Carol

Hi--I think that the quality of rehab would be best off at the nursing home-however there are many factors that you may want to address,..such as where would she have a better quality of life? As Crole has said, this decision is up to you and that of your family-and the POA for your Mom's health, if there is one.
Hap