The answer is no, the whole world is not saved.
There is something that needs to be addressed here, it comes up much too often. John 3:16 does not say the WHOLE world, it only says "the world". If you look up the meaning of the original text, you will find that it is in relation to a PART of the world and not the whole. If the opinion is that the whole world is loved by God, it is not. Only part of it is, that being those whom He has called. Just read Romans 9-11, then all the Epistles, you'll see.
That being established, back to the question. Should the whole world be saved? No. If God had not decreed that the whole world should be saved then it should not be. It has no power to make, or even believe, that decision. NO.