The melon could not fit on the hat because it was too small.
The melon could not fit on the hat because it was too big.
You can investigate this yourself at http://corpus.byu.edu/coca/ ; the first hundred results for "fit on" contain, by my eyeball estimate, more than 90 of the sense I describe, zero of the sense you insinuate, and a few spurious hits (such as "to spend as it sees fit on government services", "kept himself fit on a rowing machine", and my favorite, "I've worked with schools such as the Pratt Institute and FIT on developing eco-friendly vegan design programs").
There are six results, out of over 500 million words, for "fit it on", of which one matches your pattern. ("She passes it under the running tap and hikes her tank up to fit it [a strip of nylon] on around her rib cage")