Unfortunately, not really. Via the permissions module, you can only currently restrict access to view pages - but the institutions / descriptions affected by changes in permissions will still be visible where linked to other entities, and in search/browse pages.
The permissions module is also very old, and simply adding more granular permissions options to it will not likely be possible without overhauling it to improve performance and scalability. I've previously written about this at greater length in the following post:
Other development options might be to add a Publication status to archival institutions. We'd likely need to determine if such a change would also affect all linked descriptions, or have an option for this. There may also be development challenges related to permissions and timeouts with this method, but it works well for Draft descriptions, which are removed from search/browse results as well. Once again however, this is something that would require development sponsorship for Artefactual to be able to implement.