Virgin Islands Visa Waiver Act of 2019 This bill authorizes the Department of Homeland Security (DHS) to establish a visa waiver program for nationals of certain countries to enter the U.S. Virgin Islands. Countries that are member or associate members of the Caribbean Community (CARICOM), such as the Bahamas or Barbados, may be part of the visa waiver program. The government of the U.S. Virgin Islands may request additional countries be added to the program. An individual entering under the waiver program shall be a nonimmigrant visitor to the U.S. Virgin Islands for business or pleasure, and may stay for no longer than 30 days. Such an individual shall waive the right to appeal an immigration officer's determination of admissibility, and also waive the right to appeal a removal decision. DHS may implement the waiver program as to a country if it determines that it would not pose a welfare, safety, or security threat. DHS shall monitor the waiver program, and shall suspend the program as to any country if it determines that an unacceptable number of individuals from that country are remaining unlawfully or unlawfully entering other parts of the United States, seeking asylum or withholding of removal, or for other good cause.