Is it correct to write "is done fully automatically" or "is done fully automatic"?

Since you're modifying a verb (the verb to do), then you should use the adverb automatically, not automatic.

What your research shows is that correct is not always the same as usual.

Note that Google's numbers of hits are usually NOT AT ALL reliable. They have become pretty much meaningless over the years, except to establish whether a phrase is used at all, and to find pages with context to study the use of the phrase.

If I go to page 10 of the results for your first link, the number changes from 19k to 48k. If I do the same for your second link, the number of hits changes from 35k to 84 [sic]. Fully automatic doesn't even have ten pages of results. This would indicate that fully automatically was used five thousand times more often; however, Google's result numbers being as unreliable as they are, I'd not use them to support anything at all. It is better to look for well-written pages using one phrase or the other.

That said, Damcrumb is absolutely right: automatic is not an adverb, and the phrase needs to be adverbial, which makes fully automatically the right choice.

It is done fully automatically.

Although it sounds awkward, it should be fully automatically, as fully acts as a sub-modifier to the adverb automatically. If you want to use the adjective automatic, then say:

It is fully automatic.

However, I would suggest the more idiomatic construction, fully automated:

It is fully automated.

And, of course, you can replace it in all these examples with the specific action you are trying to describe.