Probably the biggest moment I've watched on TV was the "One small step for man..." moonwalk. I was a small child, but I remember it quite vividly.
Second place would probably be JFK's funeral, although since I was even younger than JFK Jr., I really didn't understand what I was watching. I remember JFK Jr and thinking he was a kid like me...and I remember the 'horsies' pulling the casket (I loved horses as a toddler - was fascinated by them). Don't remember the assassination or anything - nothing about Dallas, Oswald, or Jack Ruby. I didn't even understand what a 'President' was. I only remember JFK Jr and the pretty horses. Come to think of it, those two things might just be my earliest childhood memory.
Third would probably be 9-11.
Fourth would probably be the Munich Olympics in 1972. Both for terrorist attack on the Israeli Olympic team...and also because of Mark Spitz.
Fifth would probably be the Challenger disaster.
Sixth would be the entire Vietnam War as a whole. Not any single night...but EVERY night, Walter Cronkite reading out the casualty statistics....and then the coverage of the anti-war protests. I remember being terrified because the draft was in effect, and the war had been going on so long we all wondered if it would EVER end. Even when he was 10, I remember worrying that my brother would end up coming off one of those planes in a box from Nam in 8 years.