No. The built in player doesn't do downloads (which would also bypass the parental controls and ratings the app is intended to enforce if set). You only get the App Store to pick apps that do this (there are numerous that do this) but they store the downloads in their own "sandbox" - not the iTunes music store.
The only non-jailbroken App I know of that injects music or movies into the main "sandboxes" is the iTunes App Store app which doesn't really go for arbitrary movies or mp3.
If, however, you are jailbroken there are numerous apps that can inject music into your library.
- MewSeek allows for downloading music, editing tags, and injecting music.
- iFile now also includes an option to add music to your library if you also have Celeste.