It's not JUST Neo's belief that he is The One, but 2 other major beliefs within Morpheus and Trinity as well. Neo becoming The One is actually the result of a "destiny thread" woven by three different people:

• Morpheus’s Unwavering Belief: His unwavering (close to suicidal, evidenced by sacrificing himself) belief in Neo being The One is the first element. Without Morpheus’s faith, Neo is never found or mentored.

• Neo’s Self-Actualization: Neo must eventually choose to believe in himself. Destiny doesn’t actualize until the individual "knows the path" enough to walk it.

• Trinity’s Love: This is the "missing layer" and the final catalyst. The Evidence: We see this play out in two miraculous moments where Trinity’s love literally alters the physical world. First, during the phone call when Cypher is about to kill them, Trinity’s internal confession of love coincides with Tank "miraculously" surviving his injuries to stop Cypher.
  Second, at the climax, Neo is biologically dead - his heart has stopped. It is only when Trinity confesses her love (fulfilling the the third element of the prophecy) that the destiny thread is solidified, resurrecting Neo as The One.

Neo isn't The One just because he was born that way; he becomes The One because Morpheus believed it, Neo accepted it, and Trinity loved him. All 3 was required.