Family Medical Leave Modernization Act This bill amends the Family and Medical Leave Act of 1993 to expand the number of individuals who are permitted to take leave under that Act. The bill grants leave to private sector and federal employees to care for a domestic partner and any individual related by blood or affinity whose close association with such employees is the equivalent of a family 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 related by blood or affinity to a service member whose close association is the equivalent of a family 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.