The International Court of Justice has ordered the United States to review 51 death-penalty cases on the basis that the Mexicans convicted of murder weren't given consular assistance by the Mexican government. I don't know the details of every single case, and I do think foreign nationals arrested for crimes should be allowed access to their nation's embassy staff.
One important question is whether or not any of these death-penalty cases are actually in federal courts. The federal government doesn't execute many people, so it's likely that the majority of these convictions were in state court. It's not clear what steps the federal government could take to halt state executions, other than passing special legislation.
These convicts all have built-in appeals -- regardless of what the World Court says or does -- and they're free to raise these issues on their own standing in state courts. The idea that some group of men in robes in Brussels could impose its will on the American judicial system is ludicrous. If Mexico has a problem with the US then their ambassador should raise the issue with our State Department, not go whining to the ICJ for relief.
Here's the "Vienna Convention on Consular Relations". Go to Article 36 for the relevant passage. My general understanding is that American courts have ruled that as long as there is no demonstrable prejudice against a defendent created by a lack of access to consular officials, there's no reason for any remedy. See this ACLU brief (which I have not read in full).
In their motion for summary judgment, the defendants argued, inter alia, that the right to consular notification and access under the Vienna Convention is not equivalent to constitutional or statutory rights. Sorensen v. City of New York, Defendants’ Memorandum of Law Supporting Their Motion for Partial Summary Judgment, (98 CV 3356), June 17, 1999, at 23. Accordingly, Sorensen was required to show prejudice in order to prevail on her Vienna Convention claim. The defendants added that Sorensen could make no showing of prejudice in the case. “She cannot show that the consulate could have done anything for her that her criminal court attorney did not or could not do.” Id. at 24.