What were Lyndon B. Johnson's most significant foreign policy concerns?
I would argue that the answer is in the singular, namely Vietnam.
When Johnson became President, almost immediately American involvement in Vietnam escalated sharply. There are many theories for this, for example the CIA assassinations of President Diem and his brother were only days before Kennedy's assassination in Dallas.
Some argue the two are linked ion that the war was really about Mafia control of the heroin trade from South East Asia through American military customs.
Whatever the reason Vietnam quickly became a nightmare. Despite over 500,000 American troops in Vietnam and billions of dollars being spent, what was initially seen as being a short conflict dragged on into years. Draft replaced volunteers and American society became increasingly split.
Probably the last straw for Johnson was the Tet offensive in January 1968 when the Vietcong switched their offensive to South Vietnam's cities. Although strategically a defeat, psychologically it shocked Americans seeing in their living rooms the Vietcong inside the US embassy in Saigon and the brutal reality of a widely covered war.
As a consequence Johnson announced he would not seek or accept the nomination of his party for the forthcoming Presidential election.