No, copyleft does not fundamentally depends on strong copyright law. That is just an implementation detail. It would work just as well--better, actually--if it was enshrined in actual law. In fact, I understand this is what the movement actually wants; unfortunately,a law like this is unlikely to get through the legislative process. Happily, we can use some "legal judo" to turn copyright onto itself, creating an opt-in system to remove copyright using existing laws.

In short: copyleft only depends on copyright because it has to, and it only has to because it's a minority position. Actual GPL supporters would be perfectly happy if the equivalent of the gpl was law and normal copyright did not exist.