Can I run a bridged 4 ohm amp into a 4 ohm dual voice coil without running the sub in parallel or series safely?

The minimum impedance of a bridged 4 ohm per channel amplifier is usually 8 ohms so if the voice coils of a single 4 ohm dvc sub are wired in series the answer is yes as long as the rms power output of the amplifier does not exceed the rms power handling rating of the sub by a large amount.

UPDATE - Guess what - I found the manual online. Your amp DOES bridge to 4 ohms which is atypical for most 2-channel car amplifiers. However, if your Rockford punch sub is a P2D4 your only wiring option is series for 8 ohms, so the amp is still only going to produce 280 watts with it connected unless you get a second sub and wire the vc's series and the subs in parallel. If your sub is a P2D2 then wiring the vc's in series will provide 4 ohms impedance and the amp will operate to the full 560 watts with it.

When your 2 channel is bridged, it becomes a 4 ohm at 560 watts. With one DVC 4 ohm, your only option is in parallel for 2 ohm or in series for 8 ohm.

I believe what you are asking is can you make your dvc 4 an svc 4 (effectively) for your bridged 2 channel amp.

This will completely blow the minds of some people here, but yes, you can.

Just connect the amp to one of the coils and short the other coil. Power handling will be reduced slightly, but the sub will basically perform as expected.

Your bridged amp should not be run on a 2 ohm load, which is what you'll have wiring the dual 4 ohm voice coils in parallel. Try running the amp, not bridged, with one voice coil connected to each channel. When bridged, what is the stability of the amp? The specs for the amp will tell what impedance load it can safely handle. The quote you commented to the other guy as "560w x 1 @ 4 ohms". That suggests the bridged stability is 4 ohms, not less than that.