After the Spanish (Castille) first discovered The Caribbean (Nowday Dominican Republic, Cuba ect.) in the late 1400s / Early 1500s were they aware / did they predict that mainland North and South America existed?
I'm assuming you're referring to Colombus' first voyage, which resulted in the tentative discovery of new lands west of Europe. Colombus' own conclusion after his voyage was that he had reached islands off the coast of India, or perhaps China (Cathay), and repeatedly referred to the land masses he encountered as "las Indias", and their inhabitants as "Indios".
So no, its safe to say that Colombus himself had no inkling that he had discovered a new landmass totally unknown to his European contemporaries. He did make the assumption that there was a continental landmass beyond the islands he "discovered" but he remained convinced after his first journey that that landmass was Asia.
I wasn't until Colombus' third voyage that he encountered and described a previously unknown continental landmass (which turned out to be South America). But even then he still wrote in his reports that he assumed the landmass to be some part of Asia.
Its important to bear in mind here that the concept of "Asia" was poorly understood and loosely applied by Europeans in the late 15th, early 16th centuries. Knowledge of the spherical nature of the earth was well known among educated Europeans during Colombus' lifetime, any many who interpreted Colombus' discoveries at the time thought that he had encountered the western tip of Asia. This is reflected in the Treaty of Tordesillas, signed only two years after Colombus' first voyage divided the earth into two hemispheres, with one half for Spain & the other for Portugal.
This image depicts the geographic thinking in the decade after Colombus' first voyage. Imagine the two circles on the map forming a sphere so that the Caribbean is off the coast of Asia and you'll have an idea of how Colombus and his contemporaries thought.
European exploration progressed rapidly after Columbus, however, and by 1513 Europeans had confirmation that the Americas and Asia were two distinct continents.