If I remember correctly the Mythbusters did a segment on this very question about 2 years ago or so. They actually mounted a .22LR round in the fuse box of a vehicle and experimented with different current draws leading up to and including a dead short. They actually inserted the fuse into the holder of the old style tubular fuse block. It did indeed fire off, if memory serves.
I realize the original question involved a centerfire cartridge and not a rimfire....but in either case if enough current flows to impart enough heating, I would think something could happen with a centerfire as well.
I think it will be if current flows from one side of the primer to the other, creating a spark. My vote is that if rimfire will ignite, so will centerfire. Its still an explosive compound pressed between two metal plates. I agree with Protroubleshooter, depends on how much current.
Update: I talked with my gunsmith friend tonight and he said that an electric current will not set off a standard primer, but that there is a small rifle primer that is designed for use with a rifle with an electric ignition (gave a specific model but I forgot).
Now I guess we all know. Heat and percussion will, regular low voltage current is a no.