Because it is contained in Scripture, "Behold, I lay in Zion a chief cornerstone, chosen, and precious: He who believes in him will not be disappointed."
The particular application of "stumbling stone" as a figure of Christ is that of comparing him to a heavy stone blocking a path or road that people travel, resulting in their stumbling and falling.  Christ, as the aged Simeon prophesied, was "set for the falling and rising of many in Israel" (Luke 2:34).  People, through their pride, stumble at the lowly birth of the Saviour and at the humility of his followers, the stern morality of his teachings, and his sharp exposure of their sins.