The Constitution gives Congress the authority to ratify Treaties. Thereby, it also gives Congress the authority to abrogate them.
Short version: If Congress ratifies a treaty that restricts Constitutional right of citizens, is the treaty Constitutional? Opinions vary, but the weight of legal scholarship is no, it isn’t. The problem is that the Constitution does give special weight to treaties, calling them the “supreme law of the land.” The issue isn’t can Congress abrogate such a treaty - of course it can (as can the President). The issue is can we as citizens challenge such a treaty on Constitutional grounds.
The issue often comes up in the context of the ban on small arms that the UN is trying to push in the form of a set of treaties. Google/yahoo treaty constitution small arms UN or something similar and you’ll see what I’m talking about.
Sry, should have said “the issue I was alluding to” in the context of the tanget discussion of treaties being the “supreme law of the land” (as mentioned in the post that I was responding to). Another point was the Ukrainian law may differ, but we (I) got onto the issue vis-a-vis US law.
"This Constitution and the Laws of the United States which shall be made in pursuance thereof; and all the Treaties made, or which shall be made, under the Authority of the United States, shall be the supreme law of the land."
It's the bolded part we're talking about.
An interesting discussion of the issue can be found here: http://www.sweetliberty.org/issues/staterights/treaties.htm