I'm not sure what Bush signed, so let me cover the basics.
As Commander in Chief and the top of the military chain of command, the President can sign orders which will remain in effect until he or a new CiC says otherwise.
If this is a diplomatic agreement (which I believe is the case) then it is as meaningless as any other diplomatic maneuver, and the US can say or do something totally different tomorrow.
If the withdrawal was part of a bill duly passed by both houses of Congress, then it is a law remains in effect until it is repealed or revised.
If the signature went on a treaty... that is where things get dicy. The full process for a treaty is:
1) It is proposed.
2) A representative of the United States (often but not necessarily the President) signs it, indicating US intent to adhere to it.
3) It is passed to the Senate for consent. The consent of the Senate requires the consent of a two-thirds majority of Senators.
4) If the Senate gives its consent, the President signs it again, indicating that the treaty now has the full status of law within the United States (see
US Constitution, Article VI, para. 2.)
If this was a treaty, then was it signed in step 2 or step 4? If step 2, then it comes down to a diplomatic gesture of good will with no legal obligations or comitment.