Senator Fielding put a motion to the Senate today calling for an inquiry into how details of the phone call - in which it was alleged Mr Bush asked Mr Rudd what the G20 was - made it into The Australian newspaper.
However, the Greens refused to back the motion, with leader Bob Brown saying the matter was not of national importance and an inquiry into the phone call would fail.
"What Senator Fielding - who is quite new in the Senate - doesn't understand is that the Prime Minister can't and won't be brought before such an inquiry," Senator Brown told Alexandra Kirk on AM before the vote.
"Either the inquiry will fail, or if there was a move to force witnesses like the Prime Minister before the inquiry, we end up with a constitutional showdown between the houses, with the potential for journalists to be brought before the bar of the house and potentially sent to the dungeon."
Independent Senator Nick Xenophon agreed there are bigger issues for the Senate to deal with.
"This is a question as to whether the Prime Minister had loose lips in a conversation of probably not that much gravity in the scheme of things ... with the lame duck President of the United States," he told ABC 2 News Breakfast.
Mr Rudd has denied that Mr Bush asked him what the G20 was but he has not confirmed whether or not he or his office leaked the other details of the call.
Earlier Senator Fielding said he would still pursue the issue, despite the lack of support for his motion.
"This was a leak at the very highest level," he told ABC 2 News Breakfast.
"Until this is resolved it leaves a bad smell and a bad air over the office of the Prime Minister and that I think needs to be resolved.
"I'm not going to go away on this issue."