Even if America took Rugby or League seriously, they could do well but wouldn't 'dominate'. At least not for a decade or two.
Once again; look at the example of Olympic Wrestling:
Wrestling is a major highschool and collegiate sport in America, with huge participation, an elite coaching program and decent funding. America couldn't manage a single men's medal at this year's World's and won a grand total of one gold at the 2008 Olympics, behind the two golds won by Georgia (pop. 5 million).
...and before you say 'oh but Folkstyle be diffrent, yo!!1', no it's isn't. There are some minor rule variations in Freestyle Wrestling and that's it. With Greco you might have a point, but even that is mainly a specialisation of one aspect in Freestyle/Folkstyle wrestling (the clinch).