Family Medical Leave Modernization Act This bill expands who is permitted to take qualifying family and medical leave and provides additional leave for parents and family caregivers. The bill grants leave to private sector and federal employees to care for a domestic partner and any individual whose close association with such employees is like a family relationship, regardless of biological or legal relationship, if (1) such individual has a serious health condition, or (2) there is a qualifying exigency due to such employees' active duty in the Armed Forces. The bill entitles an employee who is a domestic partner, next of kin of a member of the Armed Forces, or any individual whose close association is like a family relationship, regardless of biological or legal relationship, to take leave to care for the service member. The bill entitles private sector and federal employees to take additional leave to participate in or attend their children's and grandchildren's school or community organization activities, meet routine family medical care needs, or care for their elderly relatives.