Does God have a gender?
I must submit this is a theological question which is a bit harder to answer.
There’s a short and quick answer to the question; gender is biological and God doesn’t have biology. According to John 4:24, God is a spirit. That means He doesn’t have a body, he doesn’t have chromosomes or genitals or any of the physical markers of gender. Physical attributes like that belong to the demigods of paganism, not to the utterly transcendent and omnipresent God of Scripture.
However, we should note in passing that this short answer isn’t disproved by the incarnation. The gender of Jesus is something that belongs to his human nature which must not be confused with his divine nature. Jesus’ masculinity no more proves that God is male than playing monopoly makes you a capitalist. Which is not to say that Jesus is simply God playing at being human, but that a person’s virtual existence in a game might have some parallels to the (incommensurable) relationship between the two realities in which the one divine Son subsists.
On the other hand, there is a long answer to the question. Gender in the Bible isn’t just about biological essences; it’s also about the structure of relationships and the story of origins. Genesis introduces us to the reality of gender through the need of the first man for a companion suitable for (and taken out of) him. Soon we see males as fathers: those who transmit their own image and likeness to their offspring (Gen 5:3). In the Old Testament fathers are the springs through which nations, blessings, curses and covenants flow out into the world.
And this (fatherhood) does have relevance for our understanding of God. If Jesus is God’s eternal son and both Scripture (eg. Heb 1:2; John 1:14) and Orthodox tradition declare that he is, then God is the original father. The human pattern wherein children participate in, and reiterate, the lives of their fathers has an original template in the life of God.
This makes the question of gender and God a bit more complicated but Christ changes everything
The differences in the ways that humans exemplify God will be with us until the end of time. We can’t wish them away in the name of greater equality and/nor should we wish to. Despite sin’s disfigurement, they remain a part of the way God acts and reveals himself in the world.
But it’s also true that these creational categories have already been eclipsed. When Paul writes that in Christ “there is not Greek and Jew, circumcised and uncircumcised, barbarian, Scythian, slave, free,” he isn’t saying that there are no differences between these groups (Paul observes that Gentiles owe Jews a debt (Rom 15:27) and that non-Jews should humbly remember that their inclusion in Israel is “unnatural” (Rom 11:17-24)), but that the differences pale compared to the fullness brought in Christ. Christ is the true image and the full revelation of God who gives fullness to those who belong to him (Col 2:9-10).
This doesn’t mean the end of gendered language. Jesus is the “Son” of God who finally reveals the “Father”. But now all God’s people, male and female, are sons in Christ (Gal 3:26-28). The term “son” which was introduced to Scripture through gendered human forms and through God’s special representatives (kings & angels) now belongs to everyone who shares in Christ. The appearance of the true Son doesn’t obliterate creational categories but relativizes and fulfills them.
In my conclusion, with the earthly appearance of Jesus, we see everything differently. Terms like man, bridegroom, father and son are found to be really about Jesus. Ultimately it’s Christ not humans (or men) in general who shows us what God is like. God is not a man; or like men; but there is one man who reveals God to us and teaches us how to be renewed in the image of our Creator (Col 3:10).