Could a tear in my labrum in my shoulder heal by itself?

No: The labrum is a structure that helps stabilize your shoulder joint as it attaches to the capsule and socket. Once torn, the fluid in your joint acts to prevent it from sticking back and the consensus is that they don't heal. The reasons to fix them are if it is associated w instability or if you have pain w/ +/- catching or mechanical symptoms. The tissue has poor blood supply.