Our constitution is unclear on who has the authority to withdraw from a treaty. There have been two cases brought up before the SCotUS concerning presidents (Carter and Bush I) withdrawing from treaties without seeking advise and consent from the Senate. In both cases, the SCotUS refused to rule on the constitutionality of this decision.
One opinion is that since the constitution requires Senate consent to make a treaty, it should require Senate consent to withdraw from a treaty, especially after the Senate ratified the treaty. Others have a different opinion that the president has the authority to unilaterally withdraw from a treaty.
One other concern is that often once treaties have been ratified by the Senate, there are often related legislative actions taken. In many cases, these include making federal law detailing the hows and whys about how the US will abide by those treaties. Depending on how the specific legislation is written, the laws may not automatically become void if the president withdraws from the treaty. In that case, we would be in the awkward position of withdrawing from the treaty (presuming that the SCotUS does not object) but still having federal laws on the books.
Now we would get into a peeing contest between congress that makes the laws and the Executive branch that implements the laws. That could get real ugly real fast.
