Yes, travel insurance is worth having for a climbing trip to Kalymnos, and the honest reason is the sport itself rather than the island generally: Kalymnos has built a genuine worldwide reputation among sport climbers for its limestone crags and bolted routes, and falls or overuse injuries on those routes are the real risk profile here, not everyday island travel.
Kalymnos itself has a basic local health center for routine issues, but anything beyond minor treatment typically means a short ferry crossing to Kos, the larger neighboring island with fuller hospital facilities, and anything more serious than that generally means onward transfer to Athens. Confirm your policy covers inter-island transfer as well as treatment on Kalymnos itself, since that first crossing to Kos is the step most trips actually need if an injury happens on a remote crag.
Sport climbing is the activity most standard travel policies handle inconsistently: some cover it as ordinary hiking or sport, while others cap coverage or require a specific climbing or via ferrata add-on. Confirm your policy names rock climbing specifically, and check whether it distinguishes bolted sport routes from multi-pitch or trad climbing, since the wording can vary meaningfully between the two, and a policy silent on climbing altogether should not be assumed to include it by default.
As an EU member state, Greece accepts EHIC or GHIC cards from eligible European visitors for state healthcare, but check whether your own card covers the specific facility you would actually use on Kalymnos or Kos, since coverage and access still vary by provider and card type, and neither card typically covers evacuation or transfer costs. Keep any clinic paperwork and a route or guidebook reference for the crag involved if you need to file a claim, and note that the main climbing season runs roughly spring and autumn, when crags are busiest and rescue response from remote sectors is best supported.