Yes and no. Here's an analogy: people have thrown rocks since the dawn of mankind, but it was only the mathematical progress in the last few hundred years that allowed us to described the laws of motion on a parabolic trajectory in a uniform gravitational field.

Do you need to know math to throw a rock? No. But knowing the math allows you to do things that you could not do otherwise, such as precisely firing high power cannons over long distances, etc.

Archimedes of Syracuse, in the year 250 B.C., in his treaty 'On Floating Bodies', formulated this principle:

Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.

Prior to Archimedes, it was definitely possible to build seafaring vessels, but it was an empirical process (*). What the principle enabled was a mathematical analysis of what keeps the vessel afloat, which then enabled more precise and complex engineering in this field. Although reaping all the benefits probably came later, closer to the modern ages.

(*) - 'empirical' doesn't mean bad. It simply means the field is not formalized, it does not have a complete theoretical basis. Working empirically can still achieve good results - up to a certain point.

Are you asking about whether Archimedes was the first to determine a mathematical law of buoyancy or whether he was the first to recognize buoyancy as a feature of matter? If the former, then Archimedes is the first person we know of to determine a mathematical relation between the buoyant force of water displaced by a submerged object and the submerged object. But buoyancy was recognized as an existing principle of matter much earlier. Aristotle reports that Thales proposed that the earth floated on water. Thales seems to have had some sort of understanding of density, but it was not mathematically laid-out (Thales is a pre-Atomic, after all), and Aristotle quibbles with it. Thales, says Aristotle, argued that the earth has the same quality as wood, that is the ability to float on water but not on air. So Thales had some concept that objects that are less dense (or as he and Aristotle would've understood it, "lighter") than others float on them, and considered this property an inherent quality of the type of matter in question, not just of wood. But he apparently could not mathematically prove how buoyant objects float, and Aristotle attempts to disprove his idea that the earth floats on an ocean by pointing out that just as wood floats on water but not on air, so does water float on top of earth but not air--therefore, Aristotle argued, earth is denser ("heavier") than water, even by Thales' own logic. Aristotle seems to have conceived of buoyancy as dependent on the size of the submerged object, since he says that earth sinks in water faster depending on its size. Aristotle also says that the Atomists, particularly Democritus and Anaxagoras, argued that the earth rested on air like a lid. Their argument was that the earth does not fall through the air but is buoyed up basically because air compresses and, being unable to move, holds up whatever is on top of it. They seem to have conceived of this as a property that also existed in water, or at least Aristotle did, as he compares this action to the water in a water-clock--the water of a water-clock, restricted in its jar, is unable to find room to exchange places with the air above it, and therefore stays at the bottom in a mass and is eventually forced through the spout at the bottom.

So it sort of depends on what you mean exactly--buoyancy was investigated and understood to a degree by different people, with varying explanations of why it worked, but Archimedes was the first to propose a mathematical law that explains buoyancy in any situation

Did the ancients have a systematic calculus that would allow them to precisely calculate the displacement tonnage and maximum cargo load of a given vessel design in salt versus fresh water? Probably not. Displacement hulls however date back at least to 5,000 BC Egypt, millennia before Archimedes articulated his principle that buoyancy is equal to the weight of the displaced liquid. Ancient shipwrights got pretty far by Archimedes' time through trial-and-error and observation.

It was certainly observed that displacement hulls could carry considerably heavier loads than rafts and barges that relied solely on the inherent buoyancy of their materials. Egyptians had already learned that by trapping air, a raft could carry heavier loads. Moreover, the effect is easily observed when loading a vessel by regarding the ship's draft line which, if ignored, will be quickly punished with sinking. Every shipbuilding culture seems to have had methods for rating ships, with rules of thumb expressing the relationship between vessel type, dimensions and cargo capacity. The Egyptians had developed basic metrics for calculating maximum load of displacement hulls. Egyptian historical records show ships being commissioned by dimension (length x breadth) and structural type, number of braces, materials, etc. required to support a given load or anticipated task. These basic methods operationally achieved approximately the same result as calculating displacement weight. Sumerians on the other hand rated their ships in terms of cargo capacity in gurs, a unit of volume, leading to mistakes one imagines if the rules for barley were used to transport denser cargo like lead. Note though, the distinction between rating your ships' cargo capacity by weight versus volume seems to express some advance in the scientific understanding of displacement.

Displacement and boats may be analogous to probability and gambling. Romans gamblers playing dice (talia) knew that rolling a Venus didn't happen very often: someone rolling 2 or 3 Venuses in row might be congratulated on his great luck or challenged as a cheat. So while the Romans didn't have a method to precisely calculate probabilities, they were nonetheless able to effectively rank outcomes through observation. A mathematical calculus of probability that would allow one to express the precise odds of rolling a Venus as 32:1, however, would wait until Cardano attempted the first systematic study of the topic (later taken up by Pascal, et al.) in the 16th century.

Classical Element theory posited that everything was made of 4 elements (earth, air, water, fire) with some adding a fifth element of aether. It was believed that things rose or sank in water due based on the interaction of these elements. It was proposed that wood floated in water due to containing enough air in its composition of earth and air.

Of course then there is the issue of metal being able to float.

Aristotelian theory of buoyancy affirmed that bodies in a fluid are supported by the resistance of the fluid to being divided by the penetrating object, just as a large piece of wood supports an axe striking it or honey supports a spoon. Of course we know this behavior in water as surface tension. They observed many phenomenon and tried their best to explain it with what was known at the time.

Even Archimedes' On Floating Bodies was not a completed understanding of bouyancy, though it was the best effort for a very long time.