Could cartilage damage in my knee stop me from horse riding?

Maybe. You can probably continue recreational riding, however, sport riding (barrel, roping, racing, etc.) requires control with the knees, which may irritate the damaged cartillage and cause pain, which could result in loss of control. Once the cartillage is repaired or cleaned, your chances of returning to sport riding is much higher.
Maybe-but treatable. "cartilage damage" most likely means articular cartilage; softening or focal cartilage damage is termed chondromalacia, can be tx'd w/ pt or sometimes surgery if focal high grade defect and/or bony malalignment; more global damage is arthritis; if you mean meniscal cartilage tear, it can be treated w/pt or surgery; see a knee specialist; in most cases you should be able to ride after treatment.