We do accept that the priest is the minister of the marriage, and it is encoded in our canon law, which requires a priest to perform the marriage when either party is an Eastern Catholic.
From the Catechism:
Although consent of the spouses is necessary, Byzantine Catholic wedding ceremonies do not traditionally include vows, as the couple are not administering the sacrament to one another. I’ll see if I can find a transcript of the ceremony online.
The Marriage Vows
The groom and bride join their right hands upon the Book of Gospels. The priest covers their hands with his stole and right hand and then says to the GROOM:
*Repeat after me:*I, N. take you N., to be my wife, and I promise to love you, to respect you, to be always faithful to you, and never to forsake you until death do us part. So help me God, one in the Holy Trinity, and all the Saints.
Then the priest says to the BRIDE:
*Repeat after me:*I, N. take you N., to be my husband, and I promise to love you, to respect you, and to give you matrimonial obedience, to be always faithful to you, and never to forsake you until death do us part. So help me God, one in the Holy Trinity, and all the Saints.
The priest blesses them with the sign of the cross, saying:
What God has joined together, + let no man put asunder.
The Crowning
The priest places a crown on the head of the groom and bride, saying for each:
The servant of God, N., is crowned in marriage for the servant of God, N., in the name of the Father, and of the Son, and of the Holy Spirit. Amen.
And blessing the couple, the priest says:
O Lord, our God, crown + them with glory and honor.
The Ritual of Marriage with Divine Liturgy (c) 1978 by Byzantine Seminary Press, Pittsburgh, PA.