By far, the most dominant foreign policy development that drove social unrest in the United States was the war in Vietnam and related military activities in neighboring Laos and Cambodia.
U.S. military involvement in Southeast Asia, also referred to as Indochina, was a direct outgrowth of both U.S. support for French efforts at reconstituting its colonial empire after World War II -- support that was part of a quid pro quo in exchange for French support for and participation in the establishment of the Western European defense architecture -- and U.S. concern about the spread of communism throughout Asia following the victory of communist forces in the Chinese civil war. With the French defeat at Dien Bien Phu by the nationalist/communist Viet Minh in 1954, debate intensified in the U.S. Congress about the U.S. role in Indochina. As "containment" of Soviet and, to a lesser extent Chinese, influence in the developing world became the basis of U.S. foreign policy, that debate began to coalesce into a growing consensus that the fall of the whole of Vietnam to communism would precipitate its spread to Laos, Cambodia, Thailand, and the rest of Southeast Asia.
[Political developments in the Southeast Asian archipelego of Indonesia are an important side note, details of which cannot be fit within this answer, but which should be considered in the broader discussion.]
Containment lead to the gradual but steady escalation of U.S. involvement in Vietnam, somewhat slowly under the Eisenhower and Kennedy Administrations, and than massively during the Johnson Administration. By the time of the approaching 1968 presidential elections, social unrest directed against the growing war in Vietnam (intertwined to a large degree with the growing civil rights movement) took on an increasingly violent tenor. This trend towards greater levels of civil disobediance reached its climax with the riots at the Democratic National Convention in Chicago.
The significance of the demonstrations at the Democratic National Convention was substantial. The demonstrations were led by the far-left end of the political spectrum against the more mainstream liberal political establishment embodied by the candidacy of Hubert Humphrey. Because the Democratic presidency of Lyndon Johnson and Vice President Humphrey was seen as responsible for the scale of escalation into what emerged as a military quamire, not only was the foreign policy consensus that had existed behind the strategy of containment beginning to crumble, but the Democratic Party was now seen as hopelessly broken. The election of Richard Nixon was the result of the social turbulence brought about by the social unrest that resulted in large part from the Vietnam War.
The social unrest that gripped the United States was matched by similar unrest in Europe, unrest that was directed against U.S. involvement in Vietnam. Social revolution transformed the political situations throughout the Western world. In the United States, the Nixon Administration's bombing campaigns against North Vietnam and Cambodia, followed by the political turbulence associated with the Watergate scandal, marked the final dissolution of the post-World War II political consensus in the United States.