Hey,
While this is absolutely true, we're not dealing with angular size at a certain distance here. We're looking at the distance required to reach a certain angular size (the angular size that starts being obscured by close waves). Since formula has it as 2arctan(0.5(x/y)), then for the angular size to be the same, if x is doubled, then y is also doubled (x is proportional to y).
(On a side note, the formula for the angular size of the object in this case is actually simply z=arctan(x/y), since we're positioned at the bottom of the object; the formula you gave is for angular diameter when the triangle formed by us and the ends of the object is isoceles; when we're in the "middle" of the object).
We can do this whole problem without trig, though --
Draw line l representing the sea. Take point P on l, and draw wave AB perpendicular to l somewhere to the right of P, with B on l. In order for ship XY to be "sunk" (with Y on l), segment XY cannot intersect PB (if it did, light could travel from XY to P without intersecting AB). Thus the horizon distance is determined by side PY of triangle PXY similar to PAB (closest XY can be without intersecting PB). Since all such triangles are similar, PY is proportional to XY.
So... what's this?
Thanks!