I think you are
The black ball in pool
Although I'm a cube, I do not have six sides;
The ball usually has the number '8' on it which is a cube
Beneath me is green, although no grass resides.
Pool tables often have a green baize (although it can be blue or red too)
My pockets are large enough — I can fall in!
Pool tables have pockets
But then you'll fall out, though at ends you might win.
If you pot the black ball before all others of your designated type, you will lose although you will need to pot at the end to win the game.
I'm solid and black, but my pupil is white;
The black ball has a small bit of white containing the number.
My letter and name is as tall as my height.
Not sure about this one, but possibly referencing the similarity between the number '8' and the letter 'B'.
My value is tied, unlike my rotund form.
The inherent value of the black ball in pool is the winning of the game and not the value '8'.
No tees are before you; no spots on me swarm,
But I despise stripes even when they're on cue.
The black ball is neither considered part of the 'spots' or 'stripes' in pool. You use a 'cue' to play the game.
My pale friend knows me, but what about you?
The cue ball
The black ball may spin a lot during play. The number '8' turned on its side is $\infty$, not easy to count.
I'm found in a pool - pool being the game rather than a swimming pool.
pass the squares I rule - not sure about this but several of the other balls have square numbers.
and a soldier must promote - an 8-ball is U.S. military jargon for a soldier often in trouble