A different USSR which wholeheartedly embraced a combination of Orthodox Christianity and Central Asian Islam might have led to a de-emphasis on religion, particularly in the US armed forces. If the rhetoric of Soviet propaganda had lent weight to the holiness of their cause, then I could see some shift - but still, "religion" is a pretty broad, ahem, church in the US.
I am concerned it's a bit ASB, but I guess you could...
If the USSR had embraced the Orthodox church pretty much straight away after the RCW (which reads a 9 on my spacebatometer) then that might be an avenue to seek to turn Greek public opinion more strongly northeastward and maybe have a very different colonels' coup (hey, the meter apparently goes up to 11!).
A Soviet Russian Orthodox Church offers the possibility of some rapprochement between the Orthodox and Catholic faiths as a transparent cover to try to get the Vatican behind a Communist Italy (which either requires "Communist" to mean something quite ahistorical, or I'll need to change the meter's scale to read in kilobats).
It's at that point, when a Red Rome offers a propaganda channel to particularly Italian- and Polish-Americans, and to Latin and South America, that whatever US administration is in power will start to re-evaluate how much place there is for God in the American Way. Or maybe create an antipope.
As an aside, Franco would be in an interesting pickle.