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Why just Spock Prime? If Spock Prime is, then New Spock would be too. After all, the two Spocks' lives only diverge after Nero arrives in 2233 and changes history. Before that point, both continuities are one. Unless you're suggesting that Spock Prime only became a reincarnation of Surak after he grew to adulthood?

But anyway, regardless. Could Spock be a reincarnation of Surak? Anything's possible, but unlikely. As we saw in Enterprise, Surak's katra (his essence, consciousness, memories, etc) is stored in a container on Vulcan. Jonathan Archer briefly became the host to Surak's katra, so during that time it could be said that he was a reincarnation of Surak (except not fully, since Surak wasn't in control of him). At the end of that arc, the katra was returned to a container.

As I said before, anything's possible, especially in Star Trek. However, I don't see why the katra would have made its way into Spock's head. Did Sarek steal it and implant it in his son's head? If so, why? How did no one ever found out? It's possible that something to that effect happened and we just haven't seen that story yet, but until then it's pretty unlikely.

A note: Spock even met Surak in one episode. Wouldn't he have recognized himself if he were Surak?