Forrest Gump is not about any single historical event, in fact, Forrest is present at almost all of the historical events that take place during his lifetime. Obviously Forrest was not present at all of the events he is in the movie in real life, he was just added into some of the scenes. It would be possible for him to have witnessed all of the events he does, but he would have had to be extremely lucky.
One event Forrest is present for is the desegregation of the University of Alabama where he was attending school at the time. It is not completely accurate, but very similar to what happened. There were most likely many students in the crowd that watched as the governor, George Wallace, blocked the doors to the school to keep the new students from registering. However he was not there when the students actually registered as it is shown in the movie; he had left when the National Guard told him to.
Another event which Forrest witnesses is Watergate. Forrest is staying in the hotel room across from the Watergate offices and can't sleep because of the flashlights shining from the office into his room. He calls the front desk and soon after the police arrive at the offices. While the security guard is actually the person who reported the suspicious activity, it is possible that someone else also witnessed this.
Forrest was also a part of the University of Alabama football team because of his running ability. During the time period he was on the team and helping them win all their games, is historically when Paul Bryant was the coach and became the coach with the most wins in college football history. While there wasn't anyone with the same running ability as Forrest actually on the team, the coach would have recruited anyone who did have that kind of ability.
Forrest is also part of a ping pong team after he recovers from being shot in the Vietnam War. Forrest is able to visit China because of his talent. While a team of ping pong players were the first Americans allowed into china since 1949, there weren't any military players on the trip. The invitation to China reopened communication between them and the United States, and led to the end of the trade embargo as well as the first presidential trip to China.
Forest is also shown in many other historical situations. As a child he starts the dance moves that contribute to Elvis' fame. When he gets older he is in the Vietnam War, and he rescues a large group of people. For his actions he is honored at the White House, where he moons the president. Forrest is involved in commercial events in history. While running across the country he is asked by a man to help him come up with an idea for a shirt, he accidentally creates the smiley face shirt. Also during his run he unintentionally comes up with the "__it Happens" bumper sticker.
With some of his money Forrest invests in "some kind of fruit company" which happens to be Apple Computers. Forrest Gump also creates the Bubba Gump Shrimp Co. to fulfill his promise to Bubba, one of the soldiers Forrest served with in the war. The company was not real at the time the movie was made, but in 1996 the Bubba Gump Shrimp Co. began as the first theme restaurant inspired by a film, and now has at least 28 locations.
One of the interesting things about the movie is how Forrest is involved in so many historical events, and doesn't realize the importance of any of them. For him it is just another day.
While it would be possible for a single person to be present at all of these events, it is extremely unlikely that anyone was, so that part of the film is not true, otherwise most of the facts are correct. The differences between the history in the film and in real life are mostly small details and are changes made to incorporate Forrest into all of the events.