Books approved by the Vatican
There is no such list.
Catholic publications receive approbation such as the Nihil Obstat and Imprimatur from the ordinary of their diocese or his appointed secretary. So this is all done at the level of particular Church and none of it goes to Rome for any kind of direct approval. I am not aware of any special program for ecclesiastical approval at the Vatican. I am not aware whether the Pope issues his own approbations as Bishop of Rome when the publication is local to that city, or if he appoints someone else for it.
In order to get a “list of approved books” you would need to go through each publisher in each diocese and make a list of everything with Nihil Obstat and Imprimatur for the last N years. That will be a Sisyphean task at best. I don’t know anyone who has undertaken it. Your best solution would be to seek reputable Catholic publishers and examine their works for these marks of approval. I am sure they will be glad to communicate with you about specific items in their catalogs.
Unfortunately, there is no such list. A list of books could be drawn up by such means as identifying saint-produced books especially admired by the Vatican when it canonized their authors and the works of Fathers and Doctors of the Church, but I do not know of such a list.
Maybe if you are looking for a certain kind of book or a list of recommended Catholic books, it would be easier to help you…