Your semicircular canals are concerned with balance (or spatial orientation). Proprioception, on the other hand, deals with where the body parts are with respect to each other and the physical world. (Some people lump the two; it's easier to think of them independently, because they really do quite different things.)
Proprioception (and closely associated kinesthetics) is possible because of nerve signals from skin, tendons, muscles, and joints. This gives us the senses of limb position and movement, the sense of tension or force, the sense of effort, the sense of gravity. Propioception alone cannot compensate for serious balance problems.
The vestibular system exists through input from three semicircular canals located in the inner ear on each side. The canals are oriented at (basically) 90° angles to each other so that we can maintain our balance no matter which position our head is in. These are also connected to one's ocular reflexes. Together they're called the vestibolo-occular system, which functions as a reflex (VOR).
Three sensory systems, vestibular, visual, and somatosensory/proprioceptive, serve our sense of balance. Each of these systems provides unique sensory information to the brain where it is integrated (often in complex ways) to produce a final motor output to muscle groups for maintaining gaze, head and body posture, and controlling static and dynamic balance.
Without these systems, we would be completely unable to get or stay in a standing position. We certainly could not walk.
An analogy would be to get strapped into one of those spinning seats, in this case, one that rapidly spins inverting you, alternating between forwards and backwards (which would kind of include up and down) but not sideways. You would be unable to focus your eyes on anything. You would see your world spinning, and you would have lost your sense of balance. You would immediately go to the ground, even though your proprioceptors were working perfectly.
However, once you did "go to ground", say, in an elevator, you probably could tell if you were accelerating upwards or downwards. Closing your eyes would help to drown out the more unpleasant 'noise' (input).
So, long story short, the answer is yes. Likewise, you would probably still be able to touch your nose, etc. But it would not be possible to do more complex things, like walk, or even sit unless your back and sides were supported.
 Proprioception From a Spinocerebellar Perspective
 mpairments of reaching movements in patients without proprioception. I. Spatial errors
 Anatomical and Physiological Considerations in Vestibular Dysfunction and Compensation