It's a valid question. After all, he has an unshakable moral character that pretty much defines who he is. While The First Avenger didn't delve into his faith, it certainly showed him as a man who believes in standing up to evil, but not wanting to take part in anything that would be evil (as we see when Dr. Erskine asks him if he wants to kill Nazis. No, he doesn't want to kill anyone).
In The Avengers, we hear him correct Black Widow that "There's only one God ... and he doesn't dress like that."
With Age of Ultron we immediately learn that Cap holds his team to high standards of "language," which is consistent with a man of faith. Then we hear Tony Stark call him, "God's righteous man."
Maybe those reading this post can think of something in The Winter Soldier that also revealed his possible faith. Off the top of my head, I think of his devotion to his friend and the fact that he loved him despite the evil that his friend was doing. That certainly lives up to a divine love that Jesus taught (Love your enemies).
I guess this is why I have always liked Captain America. Even in the comics, he was developed as a man of faith, filled with a true righteousness, not self-righteousness. He stood up for the moral right, regardless of who it helped or chaffed.
Kudos to Marvel and Kevin Feige, Joss Whedon and the Russos for keeping him that way. I think pop culture needs to be reminded that it is still awesome to be made of stern moral fiber.
As Agent Coulson said in The Avengers, with everything going on in the world, we need a little old fashioned.