If it only took marriage to be a citizen, the population of the US would skyrocket! You have to go through a years long process to get legal US resident status (2 years with the first one, then more meetings, paperwork, money to maybe be approved for the 10 year resident status). I think it's 5-10 years before you can become a citizen.
There are a lot of players from Europe that now have legal resident status and there are a handful that have dual citizenship and have become citizens. I can think of Ewa Laurance, Kelly Fisher, Thorsten Hohmann, Karen Corr, Julie Kelly and I'm sure there are some others that have gone as far as getting citizenship. I'm not sure of the Mosconi Cup requirements about playing for one team or the other when you are a citizen of both a European nation and the USA.