There are actually a couple of overlapping settings for this song. Played in the final scene of Armageddon, as Grace marries A.J. and casts an appreciative and loving glance at her heroic but dead father (can you believe we managed to make it this far without mentioning Bruce Willis even once?), the song's most literal setting is in the final scene of the film.
But as Liv Tyler plays Grace and her actual father sings the song, "I Don't Want to Miss a Thing" could also be situated within their relationship.
Within the film, Grace gets married immediately after losing her father in a heroic battle against an earth-bound space rock. In cheesy Hollywood fashion, the song's lyrical pledge not "to miss a thing" could be applied both to the new couple and to the father and daughter. Grace's father, Harry Stamper (seriously, Brice Willis), has been blown to bits by the double-whammy of a nuclear bomb and an exploding asteroid, but we know he is still there in spirit—or maybe a trillion tiny space spirits—and not missing a thing.
In real life, Steven Tyler missed the first decade of his daughter's life, as Liv's mother chose to conceal her daughter's paternity in order to protect her from the influence of a drug-addicted rock star. But after the relationship was revealed in 1988, Liv and Steven developed a close bond. He had three other children through other relationships, but he described Liv as his "dream daughter" and has promised to make up for lost time. He said, "I love her more than anything and have a great relationship with her. We can talk about anything." (Source)