|2000 US Census Data on Self Reported Ancestry|
Others try to calculate which group most Americans today have roots in by using immigration numbers plus basic multiplication to determine their population growth. The trouble with this is that the multiplication is an estimate, and again, so many Americans today are a mixture of European ethnic groups. In the end, we can only say that it's impossible to determine with any real accuracy which single European ethnic group most Americans today are descended from. But the immigration stats are still helpful by showing us the largest European groups to legally settle in the US.
When estimating the amount of descends today from each of these groups, it's important to consider the time periods in which they immigrated. The longer an immigrant has been here, the more descendants they will likely have because they had more time to multiple. For example, the Italians may have had more immigrants than the Irish, but the largest period of Irish immigration occurred decades before the bulk of the Italians arrived so theoretically, it's possible the Irish have more descendants. Considering many Irish and Italians intermarried, it would not surprise me if they were about equal though.
With this in mind, if you look at the stats I compiled in this spreadsheet, you'll see that the Germans and British not only have the highest immigrant numbers in total, but also are the only groups to have been consistently immigrating in mass numbers since colonial times all the way up to 1969. I think it's safe to say that most Americans today have German and/or British ancestry. It would probably be impossible to determine which out of the two of them would rank higher, and many people probably also have some other groups mixed in there - personally I also have Italian and Norwegian, but not everyone else will. But it seems obvious from these stats that most Americans with European heritage have some German and/or British ancestry, even if they don't know it.