I may not be using the correct terminology; please correct me if so.
There is an existing question How to calculate the area covered by any spherical rectangle? which is similar to mine, but I am looking for the volume of the pyramid-like shape bounded by a spherical rectangle and the four planes formed by each of its sides when taken together with the center of the sphere. That is, if the corners of the spherical rectangle are labeled A, B, C, and D, and the center of the sphere is labeled S, I am looking for the volume bounded by the spherical rectangle ABCD and the four planes ABS, BCS, CDS, and DAS.
If the spherical rectangle $ABCD$ has area $S$, covering a $\frac{S}{4\pi r^2}$ fraction of the surface of the sphere, then the pyramid-like region covers the same fraction of the volume. From $\frac{S}{4\pi r^2} = \frac{V}{\frac43 \pi r^3}$ we get $V = \frac r3 \cdot S$.
The same holds for any shape on the surface of a sphere (and the spherical section we get if we extend every point of that shape to the center of the sphere).