Can the prefix a- be appended to the word schismatic to form the word aschismatic, meaning the opposite of schismatic? Both the prefix a- and the word schism(atic) seem to be of Ancient Greek origin, so from my limited understanding it appears that this could be a valid word, yet I have never seen it used.
This is the alpha privative, which is often used to negate words of Greek origin. As schism is of Greek origin, it can certainly be used; and a web search does indeed throw up occurrences. As it's only used in obscure religious contexts, Google n-grams are not really a reliable source. It is, however, exceedingly rare, and a lot of the web search hits actually result from a schismatic run together by mistake. A cursory glance at some results only shows one genuine (though rather tongue-in-cheek) usage. So it is a valid word, and one that is easy to understand due to its productive formation, but still rather obscure.
We can (and often do) communicate with others using new phrases and sentences, breaking apart other phrases we've heard at some point in our life into words, moving them around, and replacing some with others we've heard. As long as the new phrase is grammatical and sensible, it works. A phrase doesn't have to have been uttered before to be "correct".
Words are altogether different. You can only break a word into morphemes to understand those that already exist. You can't reshuffle them, even if you are following rules that make sense. Your new word is an error. If you're a celebrity, you might get others to make the same error. If they keep doing it, over time your error might become a new word, but for now it's still a mistake.
So, following on @user110518's answer. No, that particular word isn't used, so the answer to your question: "Can I add a prefix to this word and make a new word", or in the comments, "is it grammatically correct" the answer is no. It's not correct because it's not (yet) a word.