You say it spread to Europe from the U.S., but it's kind of the other way around. The whole anti-vaxx movement was—although not started—heavily popularized by Andrew Wakefield, a medical scientist who very publicly brought criticism against the MMR vaccine (with an unethical study which lied about the condition of many of his patients) about it potentially causing autism. Remember, not too many years ago being autistic was seen as something so much worse than it is. In the meantime, he was being very privately paid off to produce a study for a lawyer who wanted proof that a certain vaccine could have caused medical complications, so he could win a law suit. There was a huge vaccine scare in Europe about MMR, and eventually it spread to America. However, as the anti-MMR-vaccine idea spread, it grew to become anti-vaccine. Wakefield, now rejected from the scientific community, had little other way to stay afloat financially than by pandering to his audience, shifting his message from anti-MMR to anti-vaxx.

Relevant video: https://m.youtube.com/watch?v=8BIcAZxFfrc (seriously, great vid, please watch if you have the time)

Relevant book: https://www.amazon.com/Doctor-Who-Fooled-World-Deception/dp/1421438003 (seriously, great book, please read if you have the time)

Although if you're talking about COVID vaccine fears, I know much less, it absolutely could have started in the U.S.