First, the spelling is important here. you mean the Bibliotheca Palatina; the Biblioteca Palatina is in Parma and is completely different.

There has been some international progress in the late 20th and early 21st century as far as repatriation of cultural assets, but the world still has a long way to go in that regard. The argument for the return of the items in the "Bibliotecha Palatina" in the modern Vatican holdings is thin at best, especially considering how long ago the removal event took place. The fate of those items had already been debated/decided several times on the world stage (European stage) in pre-modern times. In 1797 was the insane Treaty of Tolentino, which empowered the French to take whatever they wanted from "Italy" as far as cultural treasures. The BP, or at least part of it, was among the vast hoard of objects selected. The BP was fractured, some of it remaining in Rome and some of it going to the Louvre. Another fragment of it probably had remained in Germany and had been lost or destroyed in the meantime. Subsequent to the treaty of Toletino, during the clean-up of the huge French imperial mess at the Congress of Vienna in 1815, France's portion of the BP (or at least a portion of that portion) went back not to the Papacy, but to Heidelberg, which in 1815 was under the Duchy of Baden. And then a year later, Pius VII "donated" a further 800 to 1000 manuscripts from the BP which had remained in the Vatican back to Heidelberg, mostly German-language texts (it seems the French didn't want those and specifically excluded them--go figure). At this point, whatever had constituted "the" Bibliotheca Palatina was very much muddled, had been split and transferred and donated and re-donated, and had lost some of its content along the way. For instance: the noteworthy Heidelberger Kapellinventar, an inventory of music (bound choirbooks, partbooks, and single leaves) which came to Rome in 1623 and eventually went back to Heidelberg in 1816 without any of the music items it catalogued, all of which are presumed lost, either at the Vatican (unlikely), in transfer to France (possible), or back in Germany when the French sacked Heidelberg in 1693 (likely, meaning the inventory traveled, but the actual items it catalogued never actually left the city). So in sum: even if the Vatican were amenable to such a repatriation (they aren't), this particular case is quite muddled.

See A. Jakubowski, State Succession in Cultural Property (Oxford 2015).