When the output stage is at it's current limit the output power will not quadruple and the output voltage will not double instead it will remain unchnaged.
When it is near it's current limit you will only get a bit of extra power, how much depends on how far it is from it's limit.

output power will only quadruple when the amplifier can at least supply double the amount of current it can under normal conditions.

Say an amplifier can provide 40W into 4Ohm it can also provide 20W into 8 Ohm.
When you bridge that amp (20W into 8 Ohm, capable of delivering 40W in 4 Ohm) the output power will quadruple into 80W in 8 Ohm but most likely will not quadruple for 4 Ohm speakers (so most likely less than 160W in 4 Ohm) when the output has a current limiter or the transformer in the amp simply cannot deliver the needed power.

For headphones the theory is the same. Most headphone amps have current limiters though.

So if you have an amp that provide 1W into 32 Ohm it can only just provide 0.1W into 300 Ohm, if the output current is NOT the limiting factor which it often is.
When you bridge that amp it can now deliver 0.4W into 300 Ohm BUT, depending on the fact that it can deliver 2W into 16 Ohm or not will quadruple the output power for the same 32 Ohm headphone or just increase it a bit. High impedance headphones, however, are almost never limited in current by the amplifier but always in voltage.
So for higher impedance headphones (>300 Ohm) the thesis would be correct the output power quadruples.