Fostering Stable Housing Opportunities Act of 2019 This bill requires public housing agencies (PHAs) that administer housing choice voucher programs to, in coordination with the Department of Housing and Urban Development, extend tenant-based assistance for youth aging out of foster care under specified circumstances. Under current law, public housing assistance is available for up to 36 months for low-income youth who are 18 to 24 years old, left foster care at age 16 or older (or will soon leave foster care), and are homeless or at risk of becoming homeless. The bill requires PHAs to extend such housing assistance for up to 24 months (subject to available funds) for youth who are (1) employed; (2) enrolled in a workforce development program; (3) pursuing postsecondary education, a high school diploma, or the equivalent; or (4) participating in a family self-sufficiency program. Youth who are parents (or have a dependent under age six), are participating in an addiction recovery program, or cannot comply due to a medical condition may receive extended housing assistance without meeting these requirements. Further, youth receiving extended housing assistance are eligible for available supportive services. PHAs also must connect such youth with other community resources and coordinate with local child welfare agencies to receive referrals for the housing assistance program. The bill does not apply to housing choice vouchers that are already in use as of the date of the bill's enactment.