No. Windows Vista, which has DirectX 10, includes an updated DirectX runtime based on the runtime in Windows XP SP2 (DirectX 9.0c) with changes to work with the new Windows Display Driver Model (WDDM) and the new audio driver stack, and with other updates in the operating system. In addition to Direct3D 9, Windows Vista supports two new interfaces when the correct video hardware and drivers are present: Direct3D9Ex and Direct3D10.
Since these new interfaces rely on the WDDM technology, they will never be available on earlier versions of Windows. All the other changes made to DirectX technologies for Windows Vista are also specific to the new version of Windows. The name DirectX 10 is misleading in that many technologies shipping in the DirectX SDK (XACT, XINPUT, D3DX) are not encompassed by this version number. So, referring to the version number of the DirectX runtime as a whole has lost much of its meaning, even for 9.0c. The DirectX Diagnostic Tool (DXdiag.exe) on Windows Vista does report DirectX 10, but this really only refers to Direct3D 10.