Does Odysseus accomplish his rescue on his own?

From a modern perspective, of course; today we believe widely that the Graeco-Roman polytheism is incorrect; therefore Odysseus did, naturally, control his own destiny.

However, following the story, there is little that Odysseus actually did himself: most of his actions he followed through as a result of a god telling him to do so. His travel was always affected by a God. However, ultimately, had the Gods not had any affiliation with the story, the outcome probably would have been the same, except that he wouldn't have been late, and a book wouldn't have been written about his journey.

In the story, however, it was the gods who ultimately affected his destiny, not the man himself.