Audio problem: At the end of the fight when Apollo tells Rocky there won't be a rematch, his lips aren't moving. Rocky's lips aren't moving either when he says "Don't want one!"
Audio problem: When Rocky is in the locker room with Spider after the first fight, Spider supposedly says (according to subtitles and closed captioning), "Hey. You really got lucky," but that's an obvious post-production overdub. His lips are saying "sometimes".
Answer: They mean the entire year, 1976, is America's birth "day," being the bicentennial of the Declaration of Independence. So the fight, taking place on January 1, is the "first sporting event on our two hundredth birthday."