No. There would have been no reason for him to die. This film was the last of the original series, but it wasn't intended to be (Tôhô tried and failed to bring him back for nearly ten years before they finally made and released
Godzilla 1985 (1985) in 1984). More to the point, Godzilla is shown to be alive in
Destroy All Monsters (1968), which chronologically comes after this movie.