Probably not. Plutarch basically states that he just wanted to escape to Cisalpine Gaul and have his fellow slaves escape to their homelands (though he did get to N Italy and eventually turned back for unknown reasons (Plutarch implies so that the ex-slaves could plunder more)). Appian and Florus state that he wanted to sack Rome, though he never actually carried through with it. No ancient sources state that he ever wanted to abolish slavery in general.

Another important point to mention is that slaves actively joined Spartacus's revolt on their own accord, rather than by Spartacus freeing them. If Spartacus had actually attempted to end slavery in total, then he would have probably worked harder to free slaves, rather than just wait for them to join him.

For the First Servile War, accounts of Siculus and Livy tell us that Eunus (the leader of the revolt) wanted to "reverse" the situation in Sicily, where he was a slave. He claimed that he wanted to be a king and possibly enslave his aristocratic masters, which doesn't lend to the idea that he wanted to end slavery altogether.

I don't know much about the Second Servile War or other slave revolts, so I can't speak to them specifically.

Sources:

Ab Urbe Condita by Livy

Civil Wars by Appian

Parallel Live (specifically Crassus and and Pompey) by Plutarch

Epitome by Florus

Bibliotheca Historica by Diodorus Siculus