I don't see why it would be a DirectX issue if you don't even get an error message. Furthermore, many people installed Civ5 and afterwards played Civ4, so if Civ5 would install a DirectX version which wasn't compatible with Civ4, then you'd see a lot more complaints here.
But you can of course uninstall DirectX, and then install an earlier version (there's got to be one on one of the Civ4 discs), and check whether that helps. It's arguably less of a hassle than uninstalling and reinstalling Civ4, but i see a better chance of success in the latter.