No, these issues are typically not handled by international standards/laws/governing bodies. Few of these even exist: academia is not organized hierarchically at the international level; for many countries it's not even organized at the national level.
In the example you cite:
Five years ago, OSU, in Columbus, opened inquiries into papers
In this case, you can see that it's the university (Ohio State University, located in Columbus, Ohio, US) doing the investigation, because the person accused is their employee. There are other Q&A on this site asking "what's the use of a university to a researcher" or some variation on that theme; well, here's one of the uses. By working at an institution that values its own reputation, researchers open themselves to investigation by the institution that they work for. This gives everyone else that works at the institution some additional credibility, in that their job is at risk if they break the rules.
Concerns about work from Croce’s laboratory first came to widespread attention in 2017, when The New York Times reported on allegations of research misconduct
The media also has a role; by reporting on these allegations, the newspaper induces others to investigate. In this case, it seems some OSU investigations occurred before the newspaper report, but the report also induced the university to investigate further. Media attention also may attract other people who were wronged by someone and perhaps have a stronger case than they realized once their accusations are pooled with similar accusations by other people.
Nature is also acting in more of a media role (rather than as an academic journal) by posting this article.
The primary bodies that deal with accusations of plagiarism in academia are:
Universities and other academic institutions, who have the power to fire researchers, deny students credit or graduation, etc.
Journals and publishers, who have the power to retract articles in their publications.
Media at large, including newspapers, who can induce others to act through public attention and embarrassment
Granting agencies, who can require academic institutions to uphold ethical standards and withhold funds from individuals or entire institutions that fail to do so. The Office of Research Integrity mentioned in the article would be an example.
Professional societies, who have influence through journals they control and any standards they set for membership. While these can be international, they are unlikely to have any legal authority. They can kick people out/prevent them from being members, prevent them from participating in their conferences/journals, and bring attention by other entities with more direct control.
Occasionally, some forms of plagiarism are also violations of legal constructs like copyright. There may be international norms, but these violations are ultimately dependent on whatever legal system has jurisdiction, typically a national or local court system. However, even these legal violations are far more likely to be resolved at another level first, such as a journal retracting a plagiarized article. Legal action is expensive and complicated and typically reserved for when other mechanisms (including threat of legal action) somehow fail.
The only international bodies I can think of are those that have a role only in setting guidelines/suggested standards; it's up to those who wish to uphold these guidelines to actually do so. An example would be COPE.