Stylistically speaking, since the neither occurs immediately after the sentence’s subject, it is in the position where one (prototypically) expects to find a verb phrase. For that reason, each of the alternatives completing the neither construction should itself be a verb phrase. So
He neither texted me nor called me
would be the preferred option. In general, forms like neither A nor B call for parallelism: whichever syntactic form A takes, B’s should be the same.
As far as grammaticality is concerned, though, the first of your three is also unobjectionable, whereas your third is out of bounds.