St. Irenaeus of Lyons writes about the fundamental conflicts of reincarnation with Christianity and with logic, in Against Heresies (Book II, Chapters
33: Against the Transmigration of Souls; and
34: Souls… Are Immortal).
Obviously he was writing against Pythagorean Greek and Gnostic ideas of reincarnation, rather than Hindu ones. (Although he would have heard of Hindu ideas; the Roman world traded with India, and it was an exotic place that Romans associated with wisdom and magic.)
There wasn’t much that Irenaeus would have been obliged to say against Stoic and Neo-Platonic ideas of reincarnation. They just believed in continual creation and destruction cycles where the universe was destroyed and then came back, during which all living beings also kept being destroyed and coming back, in exactly the same forms, living the same lives, and making exactly the same decisions. (A few Stoics opined that Bob in Universe One and Bob in Universe Two were not actually the same person, but they would necessarily look and act exactly the same, would have the same name, and would have identically appearing relatives, wife, farm, kids, etc.) Obviously Christianity taught one universe with a beginning and an end, so this “Grand Cycle” stuff couldn’t happen.
St. Patrick also got involved in teaching against reincarnation. We don’t know much about it, but we do know reincarnation and transmigration of souls was part of druidic teaching; several classical Roman writers agree on this. There’s stuff included in the St. Patrick materials that talks about how the pagan Irish believed that the souls of young children became butterflies. (Pretty typical Indo-European idea, and also found in some Greek mythology that associated the psyche with butterfies.) This still lives on in some Irish folklore associating white butterflies with recently dead babies traveling to Heaven, which is why you never mess with white butterflies.
So no, there’s nothing intrinsically “non-European” about reincarnation. It’s just not real, that’s all.