In reality, John 15:26 supports both the Eastern Orthodox and the (Western) Catholic positions, because (at least as far as the Catholic Church is concerned) both positions are valid and complementary.
(Note that Eastern Catholics—those who follow the same rites as the Eastern Orthodox but are in communion with the Bishop of Rome—continue to favor the Greek approach to Trinitarian theology, and, like the Orthodox, omit the expression “and the Son” when they recite the Creed in their liturgies, for the reasons that I will explain below.)
How the Greek and Latin Fathers understood “procession”
It is important to keep in mind that the Greek and Latin Fathers understand the concept of procession differently. This difference is at the root of the misunderstanding that resulted between the Eastern Orthodox and Catholic Churches since the Great Schism.
The Greek Fathers, starting with St. Basil the Great, all agree that the Holy Spirit has its ultimate origin (Greek: ekporeúetai) in the Father. The etymology of that term is important: it is a compound of ek, which means “out of;” and poreúomai, which means “to go” or “to come.” The compound means, therefore, “to come from” or “to come out of,” in such as way as to stress the original source.
The Latin Fathers, on the other hand, developed a Trinitiarian theology that focused on the communication of the Divine Essence from the Father to the Son, and through Him to the Spirit. That communication they called processio (using Tertullian’s terminology). Again, the formation of the term is important: in Latin, the preposition pro means “before” or “in front of;” and the verb cedo means “to go.” Procedo (from which processio is derived), therefore, means “to go forward.” Unlike ekporeúomai, the Latin term does not so clearly stress the original source.
Hence, we can say, when the Eastern Fathers consider of the Holy Spirit, they ask “Where does He come from originally or ultimately (ekporeúetai)?” The answer is “the Father.”
The Western Fathers, however, ask “Whom is the Holy Spirit in front of (i.e, from Whom does he proceed)?” The answer is “both Father and Son.”
The translation of John 15:26 and resulting confusion
There is no fundamental contradiction between the two approaches. What happened historically, however, is that the Latin translations (both the Vetus Latina and St. Jerome’s Vulgate) translated John 15:26 as follows:
τὸ πνεῦμα τῆς ἀληθείας ὃ παρὰ τοῦ πατρὸς ἐκπορεύεται
Spiritum veritatis qui a Patre procedit
(The Spirit of truth, who proceeds from the Father)
Notice that the term ekporeúetai has been translated with procedit—a legitimate translation, if we take the terms in their common, everyday meanings. However, in the technical theological vocabulary of Trinitarian theology, they have very different meanings.
This translation later found its way into the Latin versions of the Nicene-Constantinopolitan Creed, a historical accident that caused a false equivalence to be established between the Greek ekpóreusis and the Latin processio, and, as the centuries progressed, increased difficulty for Greek and Latin-speaking Christians to understand one another.
As late as the seventh century, St. Maximus the Confessor was aware of this very difficulty, and attempted to “translate” between the Latin-speaking Roman Church and the Greek-speaking church of Constantinople. He writes, regarding this problem, in an his Epistle to Marianus:
I have asked the Romans to translate what is peculiar to them in such a way that any obscurities that may result from it will be avoided (Epist. ad Marinum, PG 91, 136 C).
Maximus is referring to expressions that can be traced as far back as St. Ambrose, who, in his treatise on the Holy Spirit written in 381 (the same year as the First Council of Constantinople which solemnly defined the divinity of the Holy Spirit) states
Spiritus quoque sanctus cum procedit a Patre et Filio, non separatur a Patre, non separatur a Filio.
Also, the Holy Spirit, when He proceeds from the Father and the Son, is not separated from the Father, nor is He separated from the Son (De Spiritu Sancto I, 11; PL 16, 120).
This affirmation was not controvertial in Ambrose’ day, because Ambrose’s counterparts in Constantinople understood that he was using processio in the Latin sense. Indeed not affirming this would result in a heresy of subordinationism (since the Father would have something that the Son does not).
On the other hand, none of the Greek Fathers would ever have claimed that the Holy Spirit ekporeúetai from the Son—since that would imply that the Son has an origin independent from the Father.
What does it mean for Jesus to “send” the Holy Spirit?
Both the Greek and the Latin Fathers would agree that the Son has the power to communicate His Essence to the Spirit. When Jesus in John 15:26 says that he will send the other Comfortor or Paracelete (i.e., the Holy Spirit), he is referring principally to the temporal mission of the Holy Spirit (the effects that the Holy Spirit in time and history: e.g., when He sanctifies or “divinizes” the faithful and gives them His grace).
However, Jesus would not have the authrity to send the Holy Spirit into the world if the Holy Spirit did not proceed ontologically through Him. This fact no one disputes, neither Eastern nor Western.
The question is what to call the relationship between Son and Holy Spirit. The Eastern Fathers give it a special name, distinct from ekpóreusis: namely proienai. This relationship could not be called ekpóreusis, because such would imply that the Son is the ultimate origin of the Spirit—clearly a heresy.
The Western Fathers, since their notion of processio was generic enough to support it, applied it to the procession of the Holy Spirit both from the Father and from the Son. In so doing they were not claiming that the Son has an origin independent from the Father, because that idea is not contained in the notion of processio.
Both the Greek and Latin approaches to the processon of the Holy Spirit are valid and complementary.
John 15:26 in the original Greek most directly supports the Greek approach, but it does not contradict the Latin approach. The Latin translation of John 15:26, which somewhat inaccurately translated ekporeúetai with procedit, was naturally interpreted in the West in the light of the Latin concept of processio. The translation of the Nicene-Constantinopolitan Creed into Latin eventually led Greek and Latin-speaking Christians to misunderstand one another and—when the political situation between the Eastern and Western Churchers grew more tense—even to accuse each other of heresy.
However (at least as far as the Catholic Church is concerned), neither side is guilty of heresy. It is just that they mean slightly different things by “procession.”
(Source for these reflections: The Greek and Latin Traditions regarding the Procesison of the Holy Spirit; see also a version with the full text in electronic form.)