That is completely incorrect by a full 180 degrees. It is missiles by far and large that spurred ship size.
Ship size is inter-related to many things.
But specifically with regards to missiles, you have the size of the missiles AND the number of missiles required on a ship.
So yes, the size of the missiles required does have an impact.
But the number of missiles per ship is directly affected by the targeting requirements for each missile.
So in the first incarnation of AEGIS+Mk41 VLS, we saw the Ticonderoga with 128 cells coupled with a really expensive SPY-1 radar at the time.
Afterwards, they realised that they put in too many cells for the targeting systems to effectively use.
So there was a decision to reduce the number of VLS cells to 96cells with the following Arleigh Burke design
But both the Ticonderoga and Arleigh Burke are the same tonnage.
And given that we are entering the age of the battle network and offboard targeting, what matters is calculating the optimal number of ships/platforms required, because the missile size is fixed. Eg.
10 ships x 48cells
or
5 ships x96cells