In the context of spherical codes, this is an instance of Rankin's bound. The largest minimum distance between $m$ points on the $(n-1)$-sphere is achieved by:

(1) The vertices of a $(m-1)$-simplex inscribed in a great $(m-2)$-sphere when $m\le n+1$.

(2) Any $m$ of the $2n$ vertices of a cross-polytope inscribed in the sphere, if $n+1<m\le 2n$.

Therefore, any $n+2$ points will have to have at least one pair at a right angle or smaller. I am sure this is in SPLAG, but I don't have it in front of me at the moment. If a reference hasn't turned up by tomorrow, I will add it when I get into the office.

In the context of spherical codes, this is an instance of Rankin's bound. The largest minimum distance between $m$ points on the $(n-1)$-sphere is achieved by:

(1) The vertices of a $(m-1)$-simplex inscribed in a great $(m-2)$-sphere when $m\le n+1$.

(2) Any $m$ of the $2n$ vertices of a cross-polytope inscribed in the sphere, if $n+1<m\le 2n$.

Therefore, any $n+2$ points will have to have at least one pair at a right angle or smaller.

Here is the reference to the original paper of Rankin: R. A. Rankin (1955). The Closest Packing of Spherical Caps in n Dimensions. Proceedings of the Glasgow Mathematical Association, 2, pp 139-144.