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[ CNRTL : ] Empr.[unt] au lat[in] *exquartare, dér.[ivé] du lat[in] class[ique] quartus « quart ».
Wiktionary states the same etymology: how does the Numeral Adjective 4 in Latin semantically connect with the notions of removal or separation or deployment?
Building on Cerberus' answer above, I think the semantic link might be tied to the development of quartus --> quartier: a piece of land; a neighbourhood; lodgings, accommodation; an encampment of troops (see "quartier")
If the é- does denote a Latin e(x)- then perhaps we could understand écarter to mean "out of the neighbourhood/lodgings/encampment" and thus away, at a distance.
What made me think of this was the original link to écarter above as the first attestation in the entry is "s'être éloigné du gros de l'armée".
When you quarter something, you divide it into four, so that you have four separate parts. This notion of separation is what developed into the sense of enclosure or separation or removal. That's why most languages have a word like quarter, quartier, kwartier, Viertel meaning "a fourth part, a separate part of the city, a quartered-off neighbourhood".
That a fourth part should have developed this sense, and not a fifth or a sixth party, is perhaps arbitrary. But perhaps it was common to divide cities into northern, eastern, southern, and western parts? We also do that in mathematics with quadrants.
It's more likely that écarter comes from *quartare. I fact, carta < Greek charta would have given charte like in old french. Carte is a Renaissance word from the same root. Comme on a écarter et non écharter, l'origine du mot est romane, et on doit "écarter" le mot carte.
