Not necessarily. That eventually becomes true if you try to go ultra-ultra-ultra-wide, but is often not the case when comparing typical setups. Height doesn't tend to change very much with tire width, and sometimes tires can get taller as rims get wider.

A (simplistic) way to visualize what's going on is to imagine that you've got a tire that always inflates to a perfectly round profile. Suppose you've got a rim with an internal width of zero, squeezing the tire's beads together. This causes the tire to form a complete circle:



Now imagine a second rim, where the internal width is equal to twice the diameter that the tire has on the zero-width rim. Now we have this:



The tire is now half of a circle with twice the diameter than before; the circumference has doubled, but the tire only exists across half of the new circle's circumference. The radius is the same as the old circle's diameter, so the inflated height is equal.