Probably by using its superior manufacturing capability and population size to drive the invaders from its shores and back across the pacific. Then given the culture (Racial superiority, White Mans Burden, ETC) at the time the Americans would deem the Japanese people sub-human. This would lead to a great human tragedy as an ethnic cleansing campaign would ensue using Atom Bombs and probably Chemicals and Biological weapons. The Soviets would join in to grab some land and show solidarity. End Result The destruction of the Japanese people (some might survive in America), a white wash of the history books and Germany might survive till 1946.
Impact on Americas culture would be massive, collaborators would be executed, and it might be used to "justify" harsher race laws.
So be joyful "cowboy" America has never been forced into a situation like that because if you think America acts reckless now, wait until someone commits mass atrocity on our shores. after all, most Americans at that time were only one generation removed from the old world mentality
You are talking as if USA had already stolen atomic technology from the fleeing european scientists when Japan attacked Pearl Harbour. If it were so that the USA had already built the big bombs at that time, then there would not have been a pearl harbour at all.
About military myth and white supremacy, note that Japan defeated the then strongest white Russian Baltic naval duck in 1905. Japan had single handedly drove away the white humans from the asian lands. Japan single-handedly defeated the British in Hongkong, Singapore, Malay and Burma.
It defeated another white duck called France and drove it away from Indo-China including Vietnam. It drove away another smaller white duck called Netherland from Indonesia. Then it drove away the biggest white duck called United States of America from the Philippines.
This is how Japan alone defeated all the western white ELEPHANTS, who just decided to surrender to a stronger force rather than fighting like RAMBO. Ramboes do not die in the films, but the fun-loving whites die in real life because they cannot fight.
So, US cowboys could have done nothing to the Japanese settlers even if they had captured a 100, 000 sq.km of W. Coast land. Then, of course, after a few years, USA would have annihilated the settlements with atoms. But, that is another topic.