Yes, Tarzan really visits America in one of the films of the series, and he actually goes to New York.

The film is called Tarzan's New York Adventure (aka Tarzan Against the World). It was created in 1942 and became the sixth Tarzan film to feature actors Johnny Weissmuller and Maureen O'Sullivan. This was the final episode of MGM's Tarzan series. Tarzan's New York Adventure was directed by Richard Thorpe. The film included New York cityscape, as well as the customary jungle settings.

More Info: