Protecting Immigrant Gold Star and Military Families Act This bill limits the deportability of certain relatives of current and former members of the Armed Forces. Certain grounds of deportability shall not apply to a qualifying alien who is (1) the spouse, widow or widower, parent, child, or sibling of a member of the Armed Forces or a veteran; or (2) eligible for a Gold Star lapel button (for certain immediate relatives of a member of the Armed Forces who died while serving). A qualifying alien is one who (1) has not have been convicted of a felony, significant misdemeanor, or three misdemeanors; and (2) is not a threat to national security or public safety.