According to this recent Nova program on Archimedes a newly discovered palimpsest of Archimedes’ The Method talks about cutting a circular wedge into an infinite number of parts and adding together their individual volumes to measure the total volume. Which is essentially calculus in all but name.
The question then becomes why, if this manuscript was available widely, no one else ever looked at his works and tried to continue them. His name was significant enough to those after him, no matter what his contemporaries thought.
There have been several explanations of why the Method seems to have been neglected.
For a start, the assumption that it was widely available can be questioned. It was originally written as a letter to Eratosthenes and nothing is known of its readership between then and its rediscovery in 1899, other than that someone thought enough of it to make a copy in the 10th century and that someone later thought this copy unwanted. Furthermore, its style, as well as its content, stands out amongst the surviving Greek mathematical texts. Whereas the normal style is to write-up very polished versions of arguments, this is more of a “look under the hood”. Elsewhere Archimedes wrote-up similar results that have been surmised to have been obtained by the same method, but where the text gives no clues to. In presenting the polished version, he may have covered his tracks. This wasn’t necessarily to help “convince” his colleagues - it was just the standard way of doing things. The Method may thus have not been seen by contemporaries as a proper work. If many of them saw it at all.
It may even have been the case that some of them were working with similar ideas.
In his History of the Calculus (1949; Dover, 1959), Boyer’s conclusion is that even Archimedes ideas fell far short of the notion of an infinite limit. Furthermore, he suggests that the difference in the 17th century was that standards of mathematical rigour had decayed. Calculus could thus be developed in a useable form, with the nasty conceptual issues not being properly addressed until the 19th century.
If you wanted to accelerate the development of mathematics, it would probably have been more useful to give Archimedes the works of Descarte rather than Newton. Demonstrating that algebra and geometry are fundamentally identical would have had profound effects on Greek thinking. Presumably.