I have something to add.
The term more often used for centurions in the gospels, and especially in this particular episode, is the Greek calque hekatontarchēs ‘commander of a hundred’ (only Mark 15:39 uses the transliteration kentyriōn). We often assume that the man who asks Jesus to cure his servant is a Roman captain, based on the term used. But the term does not necessarily always refer to Roman centurions. For example, most of the Jewish historian Josephus’ usages of* hekatontarchēs* do refer to Roman officers, but others occur in his recounting of biblical stories, and in one case it is even used for officers of the Jewish rebels (Antiquities 6.40; 7.233, 368; 9.143, 148, 151, 156, 188; for Jewish officers; War 2.578). In addition, the term appears frequently in the Septuagint, obviously in non-Roman contexts (cf. Exodus 18:21, 25; Numbers 31:14, 48, 52, 54; Deuteronomy 1:15; 1 Chronicles 29:6 and elsewhere).
The thing is, there were probably no Roman soldiers stationed in the Galilee at the time of Jesus. For one, Herod Antipas had his own troops. (Judaea, being a Roman province directly governed by a prefect ever since AD 6, would obviously have a small number of Roman troops in it - most of whom stayed with the prefect in the gentile capital of Caesarea Maritima, far away from the Jews.) Plus, he was generally on good terms with Rome - he was, in fact, a childhood friend of Tiberius - and so would not have needed Roman soldiers. Client territories were typically a source of manpower for the Romans, not additional territory for them to garrison. Roman troops were indeed occasionally supplied to client rulers whenever they faced internal or external threats, but that was the exception, not the norm. There was no serious military threat to the Galilee during the time of Jesus, so Antipas would not have needed Roman soldiers. There was that incident with the Nabataeans during the mid-30s (Josephus, Antiquities 18.113-115), but those troubles apparently began and escalated without the Romans’ awareness - a development difficult to explain if there were Romans in the Galilee, particularly in Capernaum.
Even if one were to argue that the term hekatontarchēs reflected Roman-style military organization, that would still not show that the unit was Roman. Herod the Great had organized his troops along Roman lines (he had a multinational army composed of Jews as well as foreigners), and it is highly likely that the Roman-educated Antipas did likewise. Considering these factors, I think it’s possible that the ‘centurion’/‘hekatontarch’ Jesus met was not a Roman captain, but someone who belonged to Antipas’ army. I think the case for it could be strengthened if we compare this episode with the story in John 4:46-54, where Jesus heals the son of a royal official from Capernaum, which ultimately could have been derived from the same tradition.
As for Judaea, the Roman prefect was of equestrian rank, meaning that he did not have enough authority to command a legion. (His superior, the legate of Syria - who was of senatorial rank - commanded three or four.) Instead he commanded auxiliary troops. Now unlike legions where the members were all Roman citizens, the auxiliaries usually recuited peregrini, non-citizens, who were awarded Roman citizenship at the end of their service. Most of the ‘Roman’ soldiers in Palestine, in fact, were not from Rome or Italy, but were local recruits from gentile areas like Caesarea Maritima and Sebaste (Samaria). The prefect’s army was not too large: at the death of Agrippa in AD 44, it consisted of one ala and five cohorts - approximately three thousand men. And the bulk of that three thousand soldiers would have stayed with the prefect in Caesarea Maritima, with the rest being spread thinly throughout various garrisons across the province. All in all, the Roman presence in Palestine is actually quite small. (Probably none in the Galilee, only a little in Judaea.) Troops were not regularly stationed there, so that when big trouble broke out, the authorities needed to call on the Syrian legate to send his legions. Sometimes though, other troops might have passed through the area or were temporarily stationed there (the Italian Cohort in Acts is a possible example).