Number of fixed points of an involution

Indeed, the number of fixed points is divisible by $2^{\dim X}$. This is actually an old result of Conner and Floyd, Periodic maps which preserve a complex structure, Bull. Amer. Math. Soc. 70, no. 4 (1964), 574-579. As @SashaP mentions, Atiyah and Bott observed that it is also a consequence of the holomorphic Lefschetz formula (Notes on the Lefschetz fixed point theorem for elliptic complexes, Matematika 10, no. 4 (1966), 101-139).

This seems to be a deceptively simple statement.

A proof not based on blow-ups, and involving instead a construction originally used by Rost in the context of the degree formula, can be found in of O. Haution's paper

"Diagonalizable $p$-groups cannot fix exactly one point on projective varieties", arXiv:1612.07663,

to appear in the Journal of Algebraic Geometry. See in particular the Theorem in the introduction.

It works on every algebraically closed field of characteristic different from $2$.