To resubmit my answer from your previous thread
This answer isn't as substantive as you're probably looking for, but part of the reason is just because they couldn't. Ship building technology (think of Galleys) made ships that were mostly rowed and were designed for shallow, relatively calm seas and coastal regions, not for trans-atlantic voyages. It wasn't until the introduction of ships like the Carrack and Caravel that that kind of transit and exploration was even possible. And to your second question, I believe the answer is "yes." Christopher Columbus wanted to find a way to get to India that didn't involve having to go through the now-hostile Silk Road (thanks in part due to the Ottoman Empire now ruling much of it) so he thought he could simply sail around it.
A reply to /u/NateY3K
Specifically in the context of westward exploration from Europe, you may want to read a previous answer:
In short, most experts correctly estimated that China was a long ways away, but incorrectly that most of the way consists of water. Columbus instead insisted that the earth was much smaller than the known estimates of the time, and that China (Cathay) was farther east. Up to his third voyage he insisted what he sailed to were points connected to Cathay. So in a way, two wrongs make for a fortuitous discovery.
The Portuguese, perhaps wisely, decided to look for a southern route rounding Africa instead. It wasn't clear that there would be a passage from the southern Atlantic to the Indian Ocean. That was a key discovery of their explorers.
In all this I have largely ignored the issue of building vessels suitable for fast navigation of the deep ocean. So it was a lot more than just inspiration, you had to have a workable believable plan to obtain the resources to explore.
Well, if they were smart, they would think about what would happen to them if the rest WAS water. Most sailors would be able to work out how far they could possibly hope to sail in a day, and would know about how many days' worth of food they could carry. Any crew would be leery of signing on to a voyage that would end with them starving or dying of thirst in the open ocean.
While I know this knowledge was quite present in Byzantine and later Arabic and Latin sources, I am not sure if the Vikings had this knowledge. There brief transoceanic path followed a more "natural progression" of exploration as they settled Iceland, Greenland and finally their ill-fated colony. This progression is much like the Polynesian island hopping that allowed them to make it from SE Asia all the way to Hawaii.
The "Age of Exploration" or the "Age of the Sail" fundamentally rested upon the coming together of advancements in late medieval European and Arabic ship building techniques, advanced knowledge of sail construction and usage, new navigational aids such as the magnetic compass, and perhaps the political environment that allowed Portugal, Spain, and other European powers to sponsor incredibly expensive expeditions.
As said by others, before major technological were made, even with the knowledge of the earth's size, going too far from shore was almost suicidal, and perhaps there are many earlier attempts to cross that have been simply gone unrecorded.
As some other users said they couldn't, but I would add, why would they?
While it's true that the parthian empire (and later sassanid) jealously protected their exlusive in the immensely lucrative silk road between China and the Roman empire, no one really looked to expand beyond their controlled borders, the Roman empire maintained its borders for roughly 4 centuries, other than the conquest of Dacia byTrajan there were minimal territorial changes.
If you consider earlier, the greeks attempted colonisation of southern italy and stopped there, simply put, things had not reached that kind of scale before the Roman empire, although there are some theories that the Carthaginians reached south africa and maybe even beyond in their explorations and that the Greeks knew about the british isles (Albion) long ago.