Roughly thirty kilometres of Cozumel's Caribbean coast has almost nothing on it. No hotels, no piers, no mains electricity to speak of, no towns. A single road runs down the windward side from the transversal highway in the north to the entrance of Punta Sur at the southern tip, and along its length there are a handful of palapa bars with sand floors and a cash box, a surf break, a limestone arch, and beach after empty beach with the open Atlantic swell coming straight in. On an island that receives millions of cruise passengers a year, this half of it is essentially empty on a Tuesday.
It is also mostly not safe for swimming, and that has to be the first thing said rather than a footnote. The west coast is calm because the island shelters it and a barrier reef breaks whatever swell gets through. The east coast has neither. There is no protecting reef, the water shelves steeply, and the currents and undertow along this shore are genuinely dangerous. There are no lifeguards. People drown here, and they are almost always visitors who assumed that because the Caribbean was calm on the other side of the island it would be calm on this one. A small number of coves are sheltered by rock formations and are swimmable in settled conditions. Everything else is for looking at, wading ankle-deep in, and photographing.
Once that is understood, the east side is the best half-day on Cozumel. The drive itself is the attraction: iron-grey limestone, scrub thorn forest, iguanas the size of cats on the verge, and a horizon with nothing between you and Cuba. The bars — Coconuts up on the bluff, Mezcalitos and the others near the junction, Punta Morena further south — serve cold beer and fried fish with a view no beach club on the west coast can match, and they take cash only. Between May and roughly October these beaches are sea turtle nesting grounds, which is why the development pressure has been held off and why you should stay off marked nests.
For cruise passengers this is a real decision rather than an obvious yes. The full island loop takes three to four hours with stops, which fits a long call with an organised jeep tour and does not fit a short one or an improvised plan. For anyone staying on the island overnight, it is the single strongest recommendation on this site. Rent a jeep, fill the tank in town, go clockwise, and give it a day.