Dear Jane,
Keep in mind that aside from Luke/Acts all of the books of the NT were written by Jews
Dear Susan,
I know this can grow into a colossal thread derail, but do you actually believe in the traditional attributions of authorship?
It's pretty obvious that the synoptic gospels are not independent texts, and I doubt that any of them was written by the person traditionally associated with them.
(Well, the gospel of "Mark" may be an original text, but even that one was most likely written after the rise of Paul, and definitely not by John Mark the confidante of Peter.
How do I know that? Well, for starters, "Mark" makes some serious mistakes with regards to Palestinian geography:
Mark 7:31
Then he [Jesus] return from the region of Tyre, and went through Sidon to the Sea of Galilee.
Sidon is North of Tyre, the Sea of Galilee to the Southeast. And while it's possible that he's merely describing a detour in this case (akin to "going from New York to Washington D.C. by way of Boston), there's also this:
In Mark 5, Jesus crosses Lake Galilee and visits Gerasa, where he drives a horde of demons into a herd of pigs (implausibly numbering no less than 2,000 animals). These pigs then rush to the lake and drown themselves.
Here's the thing, though: Gerasa does not and did not lie on the shore of the lake. It's more than 30 miles (or fifty kilometers) from that city to the Sea of Galilee.
That's BASIC geography, and certainly something that any native of that land would be aware of.
But maybe he was a diasporic Jew, you say? Well, he didn't know Palestinian custom too well, either:
Mark 10:11-12
He [Jesus] answered, "Anyone who divorces his wife and marries another woman commits adultery against her. And if she divorces her husband and marries another man, she commits adultery."
This suggests that women were allowed to divorce their husbands. They were not, and any Jew should know that.
Once more into the breach:
Mark 7:3-4
For the Pharisees and all the Jews, do not eat unless they thoroughly wash their hands, thus observing the tradition of their elders; and they do not eat anything from the market unless they wash it; and there are also many other traditions that they observe, the washing of cups, pots, and bronze kettles.
Evidence in the Talmud shows that this ritual cleansing was ONLY obligatory for priests - and while some Pharisees might have engaged in this ritual voluntarily, this was certainly not true of "all the Jews" at the time.
In short: whoever "Mark" was, he was not a Jew, and not an inhabitant of Palestine.