Immediately after WWII, Germany, Japan and Italy lost their navies, while Soviet navy was not a serious force to take into account. The armored ships remained in the navies of those who won WWII. (US maintained and deployed battleships until the end of 20s century, but their armor of course played no role). When the next naval arms race began (with Soviet Union, in the late 60th) the weapons were already very different from WWII weapons: guided missiles (and guided torpedoes). No armor can protect you from guided missiles. Especially from the nuclear ones. And all countries who had substantial navies in the 1960s were thinking of a nuclear war.
EDIT. But the most important consideration is what your potential enemy does. After WWII, the US concluded that aircraft carriers and submarines were the desirable ships, and kept some old battleships, just for the case. Soviet Union never built a battleship, but they had some foreign-built. They were working on how to destroy the US carriers and submarines, and decided that building battleships is not a solution. And so on.