Yes, indeed, Birkhoff-James orthogonality is an orthogonality in the sense of Rätz. A proof appears on p.36 of

> J. Rätz, On orthogonally additive mappings, Aequations Math. 28
> (1985), 35-49.

The argument there is a refinement of the proof given on p.188 of 

> K. Sundaresan, Orthogonality and nonlinear functionals on Banach
> spaces. Proc. Amer. Math. Soc. 34 (1972), 187-190.

 

Since it is a non trivial proof, I will not give more details about it in this post.