I would argue that the statement that we can actually justify is "God helps those who believe in him." The major characteristic of Father Gilligan in this poem is not belief in himself, but rather a strong belief in and dedication to God.
In the poem, God helps Father Gilligan because Gilligan has worn himself out in God's service. Gilligan does not show any real belief in himself. Instead, he believes strongly in doing God's work. We can see this quite clearly in the following stanza from the poem:
"I have no rest, nor joy, nor peace
For people die and die";
And after cried he, "God forgive!
My body spake, not I!"
When Gilligan says this, he is revealing a deep faith in God, not a belief in himself. He then drops to his knees in prayer but falls asleep. This is when God helps him out by presumably sending an angel that looked like him to the dying man.
From this, you can see that God's help comes about not because Gilligan believes in himself, but because Gilligan is dedicated to God.