Snopes says it is "false," but doesn't really lay out a strong case for such a definitive judgment. While they do raise some cause for concern or doubt, nothing in the Snopes article conclusively disproves the alleged miraculous nature of the spiral staircase at Lorreto, NM.
Again, even though Snopes says this is "false," there really isn't ANY evidence to disprove the miraculous explanation to this staircase. Yes, they explain some of the technical aspects, but not who this carpenter was and the fact he did it alone and the matter of no visible means of support (the side support they mention does not give vertical support, only horizontal stability). The "honest" assessment from Snopes SHOULD be that it is "inconclusive" to prove one way or the other.