Are there any records that indicate reactions to such news?
Edit: Sorry for the vague question and thank you so much for the wide array of answers
The fact that, until 2012, there was apparently an island located a few hundred miles from the Australian coast that turned out to not actually exist (despite satellite mapping and bathymetric data) suggests that we might not actually have reached this point yet.
This is the 'obituary' from the American Physical Union: http://onlinelibrary.wiley.com/doi/10.1002/2013EO150001/epdf
What do you mean by your question exactly? That all lands had been mapped? To answer that, I suppose when satellites were first used to definitively draw maps (or rather use massive pictures in lieu of maps). There are still areas of our world that we have not exhaustively explored, not even considering what little amount we know of the oceans.
However, if you're asking who the first people that had a general idea of all "four corners" of the Earth, it would be less of a concrete answer. But to try to answer, I would say the Vikings. They traded with nearly all other contemporary civilizations and would have a good idea of where their goods were coming from.
The Wikipedia page also has some decent sources on westward colonization. Also, there is a crude map depicting what may have been the Norse view of the world, although I have no idea how the author came to that conclusion. Using their knowledge gained from westward exploration and eastward trading, the Vikings would have formed a nearly complete (no knowledge of the Pacific whatsoever) image of the landmasses of the world.
If you mean a general understanding of the oceans and landmasses, then it would likely be the Portuguese in their heady days of pioneering oceangoers. for example the Magellan/Elcano expedition was the first to circumnavigate the globe in 1521.
Now obviously we are to this day discovering finer details about the earth, so strictly speaking it still hasn't been mapped completely.
The way I interpret your question, I put forth that when Norwegian Roald Amundsen's expedition was the first to successfully navigate through the northwest passage in 1906, it removed one of the last significant blank spots from world maps.