Just looking at the stats on Wikipedia that Iska linked us to, the average 20+, White, American man is 5'10.5'' while the average Urban, Chinese, 17 y.o. is 5'7'' and the average Rural, Chinese, 17 y.o. is 5'5.5''. So, even at an age where they might not be finished growing, young, urban Chinese men are only 3 inches shorter than white American men of all ages, on average. I included the rural stats because IMO that shows that diet makes a considerable difference.
So I'm guessing that Asian American men can't be any more than 3 inches shorter than White American men, and, when you account for diet, I'll bet the difference is even smaller than that.
I doubt that Asian American men are considerably shorter than their White counterparts. Although, in my own personal experience, I've seen less height variety in Asian men and I've never met one as tall as myself. But I have't met any that struck me as particularly "short".