The one-arm push-up is my single favorite whole-body pushing exercise. In fact, it's what's dethroned the bench press as my new king of upper-body pushing exercises and it's what I use to gauge an athlete's pushing strength.
Generally, if you're a big dude that can crank out 4-6 plus full range one-arm push-ups, you're a stud. For smaller guys, you've got to do at least 7-10 to get your man card.
Although the one-arm pus- up isn't performed from a standing position, it still heavily involves the core, hips, and lower body. When performing one-armed push-ups, you're promoting unilateral strength, which builds better left/right side muscle balance and brings in more core activation.
But you're also training the "serape effect," which is the force production relationship between your shoulder and opposite hip (through the torso) that's responsible for actions like running, throwing, punching, etc.