Judaism is a mono-ethnical religion. Thus, being Jewish is a nationality and religion. Scheyer is an American and not an Israeli, of course. Should he go to Israel, he is antitled to receive the Israeli citizenship due to the Right of Return. If he choses to do so - he doesn't have to get rid of his US passport to do so and professionally it will only help him - he will be considered a domestic player and as he hasn't played on any US NT before, he will be able to play for the Israeli Senior NT, if chosen.
Catholicism, on the other hand, has absolutely nothing to do with nationality as Christianity - in this case Catholicism - is a missionary religion, while Judaism isn't. Of course, you can find states like Spain or Poland, where the vast majority (way over 95%) is Catholic but this has its roots in historical European demography and migration.
Scheyer as Bluthenthal is Jewish and is entitled to get the Israeli passport instantly. Sharp, on the other hand, is not Jewish but an Israeli citizen after he spent so many years (16 now, I think) in the country with all kinds of obligations for an Israeli citizen. I think, he got his passport after 5 or 7 years in the country, where he has married and got children.