John 15:26 seems to be the basis upon which the Eastern Orthodox Church believes the Holy Spirit proceeds from the Father alone. Here is a partial quote from an article regarding the Eastern Orthodox teaching about the Filioque, although it should be noted that it makes no reference to any biblical text:
The position of the Eastern Orthodox Church regarding the Filioque controversy is defined by the Bible, teachings of the Church Fathers, creeds and definitions of the seven Ecumenical Councils and decisions of several particular councils of the Eastern Orthodox Church.
The Eastern Orthodox interpretation of the Trinity is that the Holy Spirit originates, has his cause for existence or being (manner of existence) from the Father alone as "One God, One Father"... The position that having the creed say "the Holy Spirit which proceeds from the Father and the Son", does not mean that the Holy Spirit now has two origins, is the position the West took at the Council of Florence, as the Council declared the Holy Spirit "has His essence and His subsistent being from the Father together with the Son, and proceeds from both eternally as from one principle and a single spiration.” https://en.wikipedia.org/wiki/Eastern_Orthodox_teaching_regarding_the_Filioque
Although there has never been a specific conciliar statement in the Orthodox Church which defined the filioque as heresy, it was denounced as heretical by the 1848 Encyclical of the Eastern Patriarchs. Here is an extract from a useful article that gives objections to the filioque on doctrinal grounds:
It is contrary to Scripture, particularly in John 15:26: "But when the Helper comes, whom I shall send to you from the Father, the Spirit of truth who proceeds from the Father, He will testify of Me." Thus, Christ never describes the Holy Spirit as proceeding from himself, but only mentions the Spirit's procession in terms of the Father.
The justifications for including the filioque in the Creed—bolstering the divinity of the Son and emphasizing the unity of the Trinity—are redundant, given the original wording of the Creed. That is, the Son already is described as "light of light, very God of very God," and so forth. The Spirit also "with the Father and Son together is worshiped and glorified." Additionally, the Creed itself begins with a statement of belief in "one God."
The filioque distorts Orthodox Triadology by making the Spirit a subordinate member of the Trinity. Traditional Triadology consists in the notion that for any given trait, it must be either common to all Persons of the Trinity or unique to one of them. Thus, Fatherhood is unique to the Father, while begottenness is unique to the Son, and procession unique to the Spirit. Godhood, however, is common to all, as is eternality, uncreatedness, and so forth. Positing that something can be shared by two Persons (i.e., being the source of the Spirit's procession) but not the other is to elevate those two Persons at the expense of the other. Thus, the balance of unity and diversity is destroyed.
Given the previous objection, the repercussions to the acceptance of the filioque into church life are potentially massive. Because how we relate to God is significantly affected by what we believe about him, false beliefs lead to damaging spirituality. One objection often raised about Filioquist theology is that it undermines the role of the Holy Spirit in the Church. Thus, with his role being denigrated, his traditional ministries are effaced or replaced. The Church's unity becomes dependent on an office, spirituality becomes adherence to the letter of the law rather than its spirit, sacraments come to be understood in terms of validity, and a spirit of legalism prevails. https://orthodoxwiki.org/Filioque#Objections_on_doctrinal_grounds
Another source of information I found explains the background and history to the schism and made this observation:
Finally, the theology of rationalistic Western Scholasticism predominated among the Latin theologians and bishops and so obscured the biblical, patristic perspective long advocated by the East, in which the Spirit is said to proceed "from the Father" (as in John 15:26) or, more rarely, "from the Father through the Son" (as in some of the Fathers). The Eastern bishops had not imbibed the rationalist intellectualism of the West, and so were unconvinced by the highly abstract and convoluted arguments of the Scholastics. Hence, the agreement of Florence, intellectually, represented in many respects an imposition of Scholastic theology. https://orthodoxwiki.org/Filioque#Attempted_reunions_and_the_Filioque_after_the_Schism
I found a non-Wiki source of information which may be helpful in explaining the Eastern Orthodox view regarding the "monarchy of the Father" as described by Fr. John Behr: https://solzemli.wordpress.com/2010/06/05/the-trinity-scripture-and-the-greek-fathers-by-fr-john-behr/