Did "didactic" go through Latin before arriving in English or did it come directly from Greek?

Did the word didactic go through Latin before arriving in English?

No.

How could it not have?

Some people learn Greek. ("I would make them all learn English: and then I would let the clever ones learn Latin as an honour, and Greek as a treat." — Winston Churchill).

Yet Websters says it came to English directly from Greek!

It is correct.

I think they are wrong.

Why? Can you cite it being used in English prior to Milton, and then demonstrate that this source was familiar with Latin, but not with Greek?

There is a Latin word, didacticus, which means didactic and was widely used in Latin texts.

And Danish has didaktiske, but that doesn't mean it came into English from Danish.

How can we find out what is didactic's true etymology?

It came into English as a coinage of John Milton's (and a true coinage, as διδακτικός means "apt at teaching" rather than "a didactic author or treatise", as he used it—the adjective sense coming later).

Milton knew Greek, he wrote in it and about it (in particular his idea Christian tragedy should combine the legacies of Greek Tragedy and Hebrew Scripture).

With his strong command of both Latin and Greek, in turning to them for a word, he would have known διδακτικός and needed no intermediary.

The Romans took over a lot of Greek words, especially in the field of science. didaktikós is originally a Greek word, when you find it in the Latin dictionary then it is the Greek word in latinized form. I think you may really ask whether etymological dictionaries are right in saying "directly from Greek". But I can't verify your statement that the word didacticus was widely used in Latin texts. - By the way, I can't find didacticus in my Latin dictionaries or online dictionaries either.

Webster has the support of the OED which shows 'didactic' as having been a relatively modern entrant into English from Greek. The earliest recorded use is 1644.

According to Etymoline it came to English from Greek via the French word didactique. But this is clearly a relatively modern acquisition. There is no suggestion that it was part of Norman French.

The OP mentioned a Latin equivalent. Is this from medieval Latin, or classical Latin?