No, Archimedes, and ancient Greeks generally, did not see fractions as numbers, and they did not use fractions as we use them today, they did not use them at all. What they used was ratios of magnitudes. Despite some superficial similarities, ratios were not fractions, and they were not single entities, numbers or otherwise, they were relations between magnitudes, numbers, lines, areas, volumes, etc. Both relata were preserved in the ratio, 7:1 was not identified with 7, inscribed sphere:cylinder was not identified with 2:3, even though Archimedes proved them equal. Equality was not identity, areas and volumes were not numbers attached to geometric objects, they were the objects.
Ratios could be compared using Eudoxus's trick, multiplied only when it made sense geometrically, the product of line ratios was an area ratio, but not added or subtracted. See What was the aftermath of the proof of irrationality of $\sqrt{2}$ for the Greeks? for more details and references, and Did Eudoxus really set out to present irrationals as Dedekind cuts? on how Eudoxian ratio theory compares to modern real numbers. Here is from Ratio and Proportion in Euclid by Madden:
"We think of a ratio as a number obtained from other numbers by division. A proportion, for us, is a statement of equality between two “ratio‐numbers”. When we write a proportion such as a/b=c/d, the letters refer to numbers, the slashes are operations on numbers and the expressions on either side of the equals sign are numbers (or at least become numbers when the numerical values of the letters are fixed). This was not the thought pattern of the ancient Greeks. When Euclid states that the ratio of A to B is the same as the ratio of C to D, the letters A, B, C and D do not refer to numbers at all, but to segments or polygonal regions or some such magnitudes. The ratio itself, according to Definition V.3, is just “a sort of relation in respect of size” between magnitudes.
If we wish to compare two magnitudes, the first thing about them that we observe is their relative size. They may be the same size, or one may be smaller than the other. If one is smaller, we acquire more information by finding out how many copies of the smaller we can fit inside the larger. We can get even more information if we look at various multiples of the larger, and for each multiple, determine how many copies of the smaller fit inside. So, a ratio is implicitly a comparison of all the potential multiples of one magnitude to all the potential multiples of the other. (Two magnitudes are incommensurable exactly when no multiple of one is ever exactly equal to any multiple of the other.) To compare two ratios, A:B and C:D, then, we ought to be prepared to compare the array of all possible (whole‐number) multiples of the first pair with the array of all possible (whole‐number) multiples of the second."
It is this sort of comparison that Archimedes uses in his Measurement of the Circle, and which is today reinterpreted as producing fractional "estimates" of $\pi$, see Who was the first to calculate $\pi$?
