Does the Bible support the concept of male spiritual headship?

Yes. No. Maybe.

This is all really tied up in the idea of Federal Headship. The most obvious Scriptural example of this is Adam and Christ.

1 Corinthians 15:22

For as in Adam all die, so in Christ all will be made alive

This is developed a lot more in the epistles, but the general idea is that Adam acted as a representative for all of mankind: a distinctly spiritual head. Christ of course acted as the same, working the other way.

So, the question is, to what extent does everyone after Adam carry on his spiritual federal headship?

In Genesis 22:18 God says to Moses

in your offspring shall all the nations of the earth be blessed, because you have obeyed me voice

Paul confirms this in Galatians 3:6-7

just as Abraham "believed God, and it was counted to him as righteousness." Know then that it is those of faith who are the sons of Abraham.

You could make similar demonstrations for prominent biblical characters such as David, or Noah. If Noah had not "by faith" (as Hebrews said) obeyed God and made the arc, his descendants would be doomed! (Mostly because they wouldn't exist)

So there seems to be a sense in which God uses the faith of a spiritual "father" (I suppose that's why we call them the patriarchs?) to direct the spiritual status of their descendants.

And of course, how can we forget the most weighty declaration of this in Exodus 20:5-6

...for I the LORD your God am a jealous God, visiting the iniquity of the fathers on the children to the third and fourth generation of those who hate me, but showing steadfast love to the thousandth generation of those who love me and keep my commandments.

Cultural Adaptation?

I suspect (perhaps this could be asked on Judaism.SE) that this idea of federal representativeness or headship became quickly ingrained in Jewish culture. As a result, all throughout the Old Testament families are put to death because of the sins of the father. In Numbers 16, Korah's entire household. In 2 Samuel 21, 7 of Saul's sons are put to death for his unresolved sins against the Gibeonites.

It seems, then, that while this might not always be a 100%, black-and-white, specific case, that in general the father acts as the representative for the family - be it a moral, spiritual, or any other -al issue.

And now?

In 1 Corinthians 7, Paul seems to generalize this even further:

If any woman has a husband who is an unbeliever, and he consents to live with her, she should not divorce him. For the unbelieving husband is made holy because of his wife, and the unbelieving wife is made holy because of her husband. Otherwise your children would be unclean, but as it is, they are holy. (vs. 13-14)

Regardless of what "made holy" means, there's a couple interpretations here that might support the federal headship theory:

1. Paul is reassuring his audience that, even if the father (the spiritual head) is an unbeliever, his wife makes the children holy in his stead
2. Both parents, whether husband or wife, contribute to the child's "holiness" and so Paul is generalizing federal headship further to both parents, not just the father.

Paedobaptists like myself would say there is definitely an element in which the parents contribute to the status of their children in the covenant community:

Acts 2:39 The promise is for you and your children...

So yes, there is definitely some biblical support for this idea and I (and I suspect most Evangelicals) would agree that to some extent the Father specifically (but also the mother) have a sort of "spiritual" leadership.

But no, it's not as if every father is acting as a second Adam, determining the righteousness of his children.

So depending on what exactly you mean: yes, no, maybe so