No. Pet birds can get along quite well even if they can't fly. In fact, many people clip their wing feathers for their own safety.
Most of the time when you think a bird's wing is broken, it is only sprained.
Either a sprain or a break can take up to 6 weeks to heal. During this time, keep the bird in a cage where the perch is only 2 or 3 inches from the floor, and the food and water are also at floor level. This way they don't have to fly or flutter and re-injure themselves. They will generally be fine in 6 weeks or less.
But if you can see that there is really a broken bone - a piece of bone sticking out, or the wing in a terribly abnormal position, you will need to take the bird to a vet or put it down.
A bird can live with only one wing, but of course it can't fly then. Parakeets are good climbers though. So it's up to you to decide.