Was the destiny of the universe determined at the Big Bang?

Yes, at least in terms of whether it would evolve into an open or closed universe.

This is determined by the expansion rate and the total average density of the sources of gravitational field at each epoch. This total average density was predetermined after the end of the Inflationary Era some 10^-34 seconds after the Big Bang. All the universe has been doing since then is coasting, and getting cooler!