What is the origin of the German word Arzt? This expression is used to designate a medical doctor. I have researched the equivalent in English (Doctor), in Italian (Dottore), and in Spanish - my native language (Doctor). There are many similarities between them, but not in the case of German.
You are looking for the etymology of the word 'Arzt'. Once you know what you're looking for, Google is a quick helper. According to wiktionary, the word dates back to ancient Greece, but some people might argue that wiktionary itself is not a good source. However, other pages such as etymologeek agree on this:
German word Arzt comes from Ancient Greek ἰατρός, Ancient Greek ἀρχι-, and later Latin archiater (Physician, especially a chief physician of a ruler.).
DWDS has a more precise derivation
Arzt m. ‘Fachmann auf dem Gebiet der Medizin, der Kranke heilt’, ahd. arzāt (9. Jh.), mhd. arzāt, arzet, mnd. arste, mnl. arsatere (nl. arts aus hd. Arzt) sind Entlehnungen aus mlat. arciater, spätlat. archiāter, archiātrus, griech. archiātrós (ἀρχιατρός) ‘Ober-, Leibarzt’, aus griech. arch(i)- (ἀρχ(ι)-) ‘Ober-, Haupt-, Erz-’ (s. ↗Architekt) und iātrós (ἰατρός) ‘(Wund)arzt’.
So Arzt probably started from Greek archiatros (main-doctor) which was adopted by Latin (archiatrus -> arciater) which was loaned by Old High German (arzat) before it finally became Arzt. So as mentioned by guidot in the comments, the word Arzt seems to relate to the main (greek archi) responsible person for healing.
The word Doktor is also used in German. However, as the English equivalent, this generally refers to a person who has obtained a PhD degree. So doctor is rather an academic title than a job description (at least in German). However, because almost every Arzt has written a "PhD"-thesis, many people use it synonymously.
In German, if you refer to the profession, you always say "Er ist Arzt". Doktor doesn't work here. In my experience, especially when talking to children, the word Doktor is used more often, sometimes in conjunction with uncle: "Da müssen wir mal zum Onkel Doktor gehen."