Veterans Visa and Protection Act of 2019 This bill implements procedures relating to the legal permanent resident status of noncitizen veterans. Specifically, the bill requires the Department of Homeland Security (DHS) to establish a program that allows (1) eligible veterans outside of the United States to be admitted as aliens lawfully admitted for permanent residence, and (2) eligible veterans in the United States to adjust their status to that of aliens lawfully admitted for permanent residence. If noncitizen aliens obtain this status, they shall be eligible for naturalization through service in the Armed Forces. Additionally, they are eligible for military and veterans benefits as if they had not been ordered removed or removed from the United States. The bill requires the Department of Justice (DOJ) to reopen removal proceedings for noncitizen veterans who are subjects of final orders of removal, make a determination with respect to whether such noncitizen veterans are eligible veterans who may be lawfully admitted for permanent residence, and adjust the status of eligible veterans. A noncitizen veteran shall be eligible if DHS or DOJ determines the noncitizen veteran was not removed or ordered removed, and is not inadmissible or deportable, based on a conviction for a crime of violence or a crime that endangers national security for which the veteran has served at least five years in prison. DHS may waive eligibility requirements for humanitarian, family unity, public interest, or exceptional military service reasons. The bill also protects noncitizen veterans or service members from being removed from the country unless the noncitizen has been convicted of a crime of violence.