If I have a tied note across a bar line, say 2 whole G notes, and I have an accidental, say a sharp, on the first G, my assumption is that the accidental applies as well to the second G because it is tied. If I wanted it to not apply, what would be the convention? Would it be simply to not have a tie note in the first place and have two separate notes, a sharped G and a G itself?
An accidental does apply across the bar line, but would need to be reapplied to any applicable notes after the second note under the tie. Ties don't apply to different notes. (You'd use a slur instead if you wanted the 1st note held until the 2nd note is attacked.)
If, say, you sharp a G and tie it across the bar line, then follow the second note under the tie with G♮, you may want to use a cautionary accidental to show that the sharp has come off. If the sharp wasn't part of the key signature, this isn't absolutely necessary, but it usually saves a bit of confusion.
The following example should make this clear.
Tied notes are, by definition, the same note carried across a bar line. When the second note is different, even though it may be the same letter name, but altered with a #, b or natural, then that note will have to be played as a separate note in the new bar. Then, the 'tie' line becomes a slur. Yes, if it's the same note, there's no need to put the # in your case. There would certainly not be a tie - there couldn't be.
This answer is direct to your question, as you stated the second G (G natural)lasted the whole bar, Which takes a different slant from Patrx2's answer, which correctly states what happens to SUBSEQUENT notes of the same name in a second bar.
Others have already stated that a tied note does not call for a repeat accidental but any following notes will. This is correct with one exception: if the tie is broken across lines, the tied note after the line break will require a repeat accidental which then counts for following notes as well.
This is one of the rare cases where line breaks affect details of notation inside of the lines.