This just popped into my head yesterday. Your mission, should you choose to accept it, is to construct a scenario where the largest religious denomination in the United States in 2009 is "Eastern Rite Catholic."
Two initial ideas come to my mind:
1) A TL where all or at least a great part of Orthodox Europe converts to Eastern Catholicism
2) Much, much greater Ukrainian and Catholic Arab immigration to the USA early on, replacing or at least overshadowing the Irish/Italian/German influxes that formed most of OTL's Catholic American population.
This would take a massive change in history. Still it might be possible. OK, firstly, we have to knock DOWN the number of Latin Rite Roman Catholics. OTL, according to
http://www.adherents.com/rel_USA.html#Pew_branches
'Roman Catholics' are almost a quarter of the population. I doubt we can get 'Greek Catholics' that high.
So... Texas never enters the Union and Irish and Poles are kept out?
The next largest single denomination seems to be Southern Baptists at ~13%, so we may want a bit more schisming among Baptists (shouldn't be hard).
The next largest group is 'Methodists' at ~8%, and if the various mergers of the 20th century hadn't happened (Methodist Episcopal, Southern Methodist and Methodist Protestant all merged in the ?1930s?; then the Methodists and EUBs merged in 1968), we could knock that figure down to 4% for the largest group there.
Still we have to get MILLIONS of Uniates (=Eastern Rite, = Greek Catholics) in the US. Hmm...
How about Russian America extends down to California and receives massive (probably forced) immigration. A goodly chunk of the exiles might well be Uniate (western) Ukrainians. Then when people get totally fed up with the Tsar's rule and the area revolts (led by Uniate Ukrainians), they also revolt against the Tsar's church (and don't doubt for a minute that the Tsar didn't have a LOT of control over the Russian Orthodox church). Eastern Rite Catholicism allows them to keep most of their worship and traditions, but frees them from the Tsar.
Then (handwaves) the area joins the US and you have millions of Uniates in OTL BC, Washington and Oregon.
Of course, if the anti-Catholic prejudice is enought to keep out Poles and Irish in the 1830s-18??, would the US really want millions of new subjects under the Pope? Well, maybe there's a huge swing in popular opinion in, say the 1880s (why?)....
But having a large 'ex-russian', partly 'ex-orthodox' population on the west coast is the only way I see to make this happen....