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.