No, Corruption / Crimson / Hallow does not spread through walls. The conversion of walls is a visual effect. Walls convert in an approximately 8 block radius from the nearest converted block. Although walls do not spread C/C/H, keep in mind that vines and thorns can spread the C/C/H.
Corruption and Hallow (and the other spreading biomes) only spread through blocks, and compatible blocks (Dirt at Normal (and at contact), Stone, Sand, Ice and Grass at Hardmode) at most. Think that, if these biomes spread through walls, you would have to demolish them.
What Blocks Can Corruption Spread Through
Corruption and Crimson can spread through sand, mud, grass, stone, pretty much every default natural block in Hardmode, but in pre-Hardmode, they only spread through grass and bushes. Converted areas will have the same background music, spawn the same enemies, and grow the same plants.