Depends on where the ship is. There's a naval historian on this sub who will be able to give a better run-down, but what I can say is that given the weather and navigation practices in the Mediterranean, it would most definitely be feasible to stop the ship unless you're in a storm, while it might be more difficult for a Spanish Galleon or Chinese Junk to do so even under the best circumstances on the high seas.

I don't know of any specific accounts of recoveries-at-sea in the early age of sail, but I can say that most preoccupations were tied to the cargo rather than the sailors, merchants or eventual passengers. An oft-repeated phrase in Venetian senatorial debates during the tumultuous early 16th century was, "A captain jettisons the cargo to save the ship." For the metaphor to be so widely used it would seem that voluntarily throwing things overboard was a more discussed topic than things and people involuntarily falling off.

Richard Walter (c.1716-95), the chaplain onboard the ship, compiled a famous account of A Voyage around the World by [Capt.] George Anson in 1740-44. In 1741 during the voyage of Anson’s ship, Centurion, a sailor was blown overboard during a storm. Unable to reverse course, the crew had to watch helplessly as he swam in vain to catch up to them. In Walter’s account:

notwithstanding the prodigious agitation of the waves, we perceived that he swam very strong, and it was with the utmost concern that we found ourselves incapable of assisting him; indeed we were the more aggrieved at his unhappy fate as we lost sight of him struggling with the waves, and conceived, from the manner in which he swam, that he might continue sensible for a considerable time longer of the horror attending his irretrievable situation.

The poet William Cowper (1731-1800) later wrote a poem (1799) about the event called “The Castaway”, according to the Oxford Eng. Dict. the first time the word “castaway” was used to mean lost at sea instead of abandoned by God.

There's not really a hard-and-fast answer to this other than it depends, most likely you would be more likely to be rescued if the day was relatively calm and the ship wasn't being forced to run before a storm or something like that.

/u/Whoosier's example from Anson's voyage is one of many examples of men being lost overboard from period accounts; the ship being unable to turn around and the efforts of the man being indicative of the genre. Ship captains weren't necessarily in the business of leaving men behind for no reason, but the safety of the entire ship and crew had to come before an individual.

I should also mention that swimming, though absolutely known from antiquity onward, was not by any means universal among sailors at the time.

In any case, though, if a man fell overboard the usual policy in the 16th-century English navy would be to throw anything that could float to the man; turn up into the wind, stopping the ship quickly; and lower a boat to try to retrieve them.

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