Not usually. One of the scare tactics you’ll hear to get you to book your shore excursions through the cruise line is that the ship will wait until all tours they schedule return before sailing but if you book your own… Well, the issue isn’t really who you use for your tours since the cruise lines also book their tours through local operators. The issue is they don’t know your whereabouts and timing when you’re not on their tours. Simple answer, make sure you do two things: give Customer Services your itinerary and contact info for the tour you are taking, and get the contact info for the ship or harbormaster so you can contact them if anything happens to provide your location and expected arrival. Think of the public relations nightmare if you called the ship to tell them you would be five minutes late and they left without you.
There are circumstances that the ship will sail without you: if you are injured and need medical attention, if you get arrested, if you wear white socks with sandals, or if the ship has no idea of your whereabouts and no way to contact you. ‘Nuff said?