I understand that Pasch's Axiom is missing from the Euclidean set of axioms, as Moritz Pasch first showed and David Hilbert told the world about. This means that there will be some theorems in the Elements which implicitly assumes that axiom. Which is the first one?
(Here Pasch's theorem is depicted, not Pasch's axiom)
Proposition 7, Book I
refers to "the same side" of a straight line.
That a straight line in a plane has two sides can be be proved using Pasch's axiom in conjunction with some order axioms (postulates) for points on a straight line (see e.g. Hilbert's Grundlagen; English translation here: http://www.gutenberg.org/ebooks/17384). Euclid's postulates don't contain any mention of a side of a straight line, so you could take the view that Pasch's axiom is somehow implicitly assumed in this proposition.
However, Pasch's axiom would not be the only candidate assumption.
Propositions 1-6 also contain assertions which don't follow from Euclid's definitions, postulates and common notions (on any reading - the fact is that the exact meaning of these is in most cases difficult to discern). Shoring up the gaps would require extra postulates. It is conceivable that Pasch's axiom could be involved as an extra postulate in some possible fixes, so I wouldn't be too dogmatic that Proposition 7 is necessarily the first.