Definition lookup in Google will show the origin or etymology of a word, usually based on root. Is there an application, web or otherwise, that returns composite words for any given root?
Yes, the Oxford English Dictionary web application provides this service.
If you look up the word composite in the OED, it reveals that its etymology is:
Etymology: < Latin compositus, past participle of compōnĕre to put together. Compare French composite (in Archit.). Introduced first in the architectural sense (2), the only one recognized by Johnson, 1755–73.
Then you go to its advanced search page and look for either of those two Latin forms in the Etymology section:
Which run produces 17 hits, of which the first few are:
However, this is behind a paywall because it’s meant for scholarly research.
You’ll need a library card from a participating library (such as from the UK) or from an institutional membership (like a university), or else an individual subscription of your own, which runs $90 per year for their “specially discounted” price or $295 without the discount.