Anvilicious: Obviously, interethnic marriage. Considering the film was made only six months after the US Supreme Court struck down laws against interracial marriage, it's also Some Anvils Need to Be Dropped.
Hilarious in Hindsight: John mentions that Joey believes that all their children will be president one day. More than 40 years later, there's a part-African president. Furthermore, John and Joey met each other in Hawaii, just like Obama's parents did.
Harsher in Hindsight: During Tracy's monologue at the end, he mentions that the love he feels for his wife "will be there if I live to be 110!" He died weeks after the movie was finished.
Spencer Tracy's closing monologue of the film, the actor's last performance.
Katharine Hepburn never watched this movie because she couldn't stand to see Tracy's last work.
Values Dissonance: Perhaps it's a measure of the progress made when you have this contrast in the audience reaction with a large portion of the population in 1967: "She's marrying a Negro; that's crazy!" to now: "She's marrying a man she only met 10 days earlier; that's crazy!"
The wide age and life experience gap between John and Joanna can also play a larger part in modern audience reaction.
Not to mention that they're going to be moving to another country, where Joanna doesn't know the language or anybody there.
Values Resonance: Although the race issues were clumsily handled for its times, and embarrassingly unwatchable today without wincing, you can understand the father's retinence about the preposterous idea of his daughter wanting to marry a man she only met 10 days ago.