Johnson could not have won in 1968. He was hated. The Vietnam War was going badly, the race riots were all across the nation’s big cities, and the Middle East was igniting.
From WIKI:
1968 presidential election
As he had served less than 24 months of President Kennedy’s term, Johnson was constitutionally permitted to run for a second full term in the 1968 presidential election under the provisions of the 22nd Amendment.[213][214] Initially, no prominent Democratic candidate was prepared to run against a sitting president of the Democratic party. Only Senator Eugene McCarthy of Minnesota challenged Johnson as an anti-war candidate in the New Hampshire primary, hoping to pressure the Democrats to oppose the Vietnam War. On March 12, McCarthy won 42 percent of the primary vote to Johnson’s 49 percent, an amazingly strong showing for such a challenger. Four days later, Sen. Robert F. Kennedy of New York entered the race. Internal polling by Johnson’s campaign in Wisconsin, the next state to hold a primary election, showed the President trailing badly. Johnson did not leave the White House to campaign.
By this time Johnson had lost control of the Democratic Party, which was splitting into four factions, each of which generally disliked the other three. The first consisted of Johnson (and Humphrey), labor unions, and local party bosses (led by Chicago Mayor Richard J. Daley). The second group consisted of students and intellectuals who were vociferously against the war and rallied behind McCarthy. The third group were Catholics, Hispanics and African Americans, who rallied behind Robert Kennedy. The fourth group were traditionally segregationist white Southerners, who rallied behind George C. Wallace and the American Independent Party. Vietnam was one of many issues that splintered the party, and Johnson could see no way to win the war[137] and no way to unite the party long enough for him to win re-election.[215]
Foreign trips of Lyndon Johnson during his presidency
In addition, although it was not made public at the time, Johnson became more worried about his failing health and was concerned that he might not live through another four-year term. Therefore, at the end of a March 31 speech, he shocked the nation when he announced he would not run for re-election by concluding with the line: “I shall not seek, and I will not accept, the nomination of my party for another term as your President.”[216] The next day, his approval ratings increased from 36% to 49%.[217]
Historians have debated the factors that led to Johnson’s surprise decision. Shesol says Johnson wanted out of the White House but also wanted vindication; when the indicators turned negative he decided to leave.[218] Gould says that Johnson had neglected the party, was hurting it by his Vietnam policies, and underestimated McCarthy’s strength until the very last minute, when it was too late for Johnson to recover.[219] Woods said Johnson realized he needed to leave in order for the nation to heal.[220] Dallek says that Johnson had no further domestic goals, and realized that his personality had eroded his popularity. His health was not good, and he was preoccupied with the Kennedy campaign; his wife was pressing for his retirement and his base of support continued to shrink. Leaving the race would allow him to pose as a peacemaker.[221] Bennett, however, says Johnson, “had been forced out of a reelection race in 1968 by outrage over his policy in Southeast Asia.[222]
Johnson did rally the party bosses and unions to give Humphrey the nomination at the 1968 Democratic National Convention. Personal correspondences between the President and some in the Republican Party suggested Johnson tacitly supported Nelson Rockefeller’s campaign. He reportedly said that if Rockefeller became the Republican nominee, he would not campaign against him (and would not campaign for Humphrey).[223] In what was termed the October surprise, Johnson announced to the nation on October 31, 1968, that he had ordered a complete cessation of “all air, naval and artillery bombardment of North Vietnam”, effective November 1, should the Hanoi Government be willing to negotiate and citing progress with the Paris peace talks. In the end, Democrats did not fully unite behind Humphrey, enabling Republican candidate Richard Nixon to win the election.
Johnson’s “health issue” - people were sick of him.
You do realize that WIKI is not exactly a neutral source, and often slants toward the opinions of the editor who has control over the subject?