Saying that 2 multi-nationals are going to duke it out in the States over the rest of the world may or may not be true - (Are a major percentage of multi-nationals headquartered in the US? If they are lodging action in the US, is it because it is easier?)
But I would tend to think that 300 Million americans and their individual actions are going to be greatly more influential on the total case load and per capita case load in the US than the approximately 70,000 multinational companies - which would be like a drop in the ocean against the individual litigations.
That said, number of cases still does not mean that people are litigious in the sense of resorting to lawsuits instead of other available means of dispute resolution. E.g., car collisions are a huge source of litigation. The US has 3.4x as many motor vehicle deaths per capita per year as the U.K. (and presumably, many more collisions too) because of our highly suburban structure. So if we have more motor vehicle/insurance lawsuits, it doesn't necessarily mean it's because we're quicker to rely on litigation. Employment litigation is another big one for regular people. Here in the US, if you think you were fired unfairly, there is no administrative agency that'll handle the dispute. If you think you were fired because of race you can complain to the EEOC, but either way the dispute will end up in court.
USA: 5805 Suits per 100k people (cf. next highest - UK/England @ 3681 per 100k) - [1]
[1] http://www.law.harvard.edu/programs/olin_center/papers/pdf/R...