How do I sew a sphere? For instance, if I wanted to make a 6 foot 3D
pacman costume for Halloween (basically a big sphere with a mouth,
eyes, legs, and arms), how would I go about sewing the sphere?

I assume I'd have to cut out curved diamond shaped sections just like
the way some world maps are flattened out.

However, what is the actual curve that I'd need for a given diameter
sphere? How would I draw the curve out on fabric so that I could cut
it properly? I could just ask this question in the math section, but
I'd still need to know how to draw the appropriate curves on fabric.

Scale it up from the two pieces of leather that cover a baseball?

Your assumption of the orange segment approach is one way of doing it.
Giving both north and south "poles" of the sphere as the beginning and
end. No matter the size of a perfect sphere, the angle remains the
same. Consider also the octagon/hexagon, beehive or soccer ball
segments. Each segment is identical, yet stretched over a given
circumference, all remains the same.

A stitch in time, saves nine.

Stan's suggestion only makes a sphere if the material is stretched,
i.e., the spherical shape provided by something else  - the yarn
filling of a baseball.

I suggest the pentagonal and hexagonal pieces used to make some soccer
balls.  This pattern allows straight seams.

Here is a picture and a pattern:
http://www.pretty-prints.com/soccer%20ball.jpg
http://www.mathcurve.com/polyedres/icosaedre_tronque/unfolded%20soccer%20ball%202.jpg

Here is another pattern requiring 12 pentagonal pieces:
http://www.jugglingdb.com/compendium/skills/equipment/making/balls/sewingpatterns.html

Of course, these will really give you a flat-sided ball shape, not a sphere.

However, if you can find a pattern for the "orange slices", that would
be simpler to sew and cut, just a bunch of identical pieces.
Also, to give it shape, you could use carbon rods along the vertical
seams.  Maybe there is a cheaper solution to this: slats of bamboo or
something. They will have to be 7 or 8 foot long, perhaps spliced to
use shorter pieces.

Good luck