Yes. Every expedition ship on this route carries a doctor, and they can prescribe, which matters most during the crossing when someone realises too late that they should have medicated. Access to the ship's medical staff is included in your fare rather than charged as an extra.
Practically, on a rough crossing the medical response is low key and routine. Sick bags appear around the ship, food can be brought to cabins, and the doctor will see people who are struggling.
What the doctor on board doesn't replace is your own preparation. Talk to your own GP before you travel about what's appropriate for you, especially if you take other medication, and bring what you plan to use rather than expecting to source it at sea.
For anything beyond routine care, you're a long way from a hospital, which is why operators require travel insurance with medical evacuation cover. The specific amount varies by operator, so check the requirement on your departure.