No, they would not. Everyone knows it's an eastern religion originating from the Middle East. Next you'll be saying Christians have not been told Jesus was not white! Like, that's also something that all Christians know.

If anything, that sort of thing has been taken too far in the other direction. It's anachronistic to cast the first century in twenty-first century racial terms in the first place. Of course nobody at the time would have called themselves "white", because that category didn't exist, and for much of the past few centuries, different varieties of racists wouldn't have counted first century Jews as "white" for various reasons.

Nevertheless, the Middle East has been populated (long before any European colonialism) by many ethnic groups that, if seen on the street in 2023 New York City, would be identified as "white". "White" people were not exclusive to northern Europe. Some even have light blond/red hair, or blue or green eyes. Did Jesus? Nobody knows, and those aren't the most common hair or eye colors there. And most people who spent 10x more outside than we do at that latitude probably had a deep tan. (Some, at least according to Xenophon, couldn't tan.)

So, it's likely that if we teleported a random first-century Jewish man into NYC, he would be considered white-passing. Looking at very insular groups, like the Samaritans, from the same area who never left probably gives us a good idea. But this is theologically totally irrelevant. I'd just say we shouldn't push "Jesus was a brown guy" any more definitively than we would push "Jesus was a white guy."