Well, that depends on what you mean by "Catholic Church"
First, I'd like to make some corrections to previous answers
a) Although, arguably, Constantine commissioned the first bibles, it is NOT true that Constantine had anything to do with the choice of Scriptures to be included, and it is CERTAIN that this topic was not addressed at the council of Nicea.
b) Martin Luther NEVER removed any Scriptures from the bible, and all of the bibles he translated and edited - including the one that was not printed until after his death - included all of the Scriptures included in modern Roman Catholic bibles, plus one extra (the Prayer of Manasseh).
OK, I'll TRY to keep this short, as I find myself answering this over and over...
The first group of Christian leaders to gather AND address this issue AND arrive at a New Testament identical with our own was the synod of Hippo in 393 C.E. The larger council of Carthage in 397 C.E. agreed with the Scriptures chosen at Hippo. This decision did NOT include the book of Baruch, but otherwise agreed completely with the modern Roman Catholic canon.
Now, why this is important...Constantine commissioned bibles, but these were Greek bibles whose contents were, as far as we can determine, the decision of the producers. Thus, we have Codices Vaticanus, Alexandrinus and Sinaiticus, all believed to be the product of Constantine's commission, each containing a complement of Scriptures different from the others. In other words, all 3 of these contain different books. (Sinaiticus contains "extra" New Testament books, Vaticanus lacks 1 & 2 Maccabees).
SO, we can argue that these Greek codices were the first bibles, but if so, there was no standard established regarding which Scriptures to include until 393 C.E. (BTW - Vaticanus and Sinaiticus are earlier than that date, Alexandrinus later).
The church at this time was what today we might consider a unified church combining all of the Oriental Orthodox churches, all of the Eastern Orthodox churches, and the Roman Catholic Church. In other words, it was a truly "catholic" church - the only Christian organization allowed to operate within the enormous region contained within the Roman empire. It was also the direct hierarchical ancestor of the churches mentioned - which means that it was no *more* "Roman Catholic" than it was "Greek Orthodox" or "Syriac Orthodox" - and no less.
NOW, next is the Latin Vulgate. THIS is important because
a) it was almost certainly the first to adopt the "standard" established at Hippo and Carthage
b) it was the first to be known as "the bible" - a term which came into use first in Latin
So, IF you consider the Latin Vulgate to be the first bible (many Christians do), there is slightly more weight on the side of Roman Catholicism, because
a) the language (Latin - Greek was most common in the East)
b) Jerome, the primary translator, seems to have been subject to the patriarch (pope, archbishop) of Rome
c) it was at the order of the patriarch of Rome that Jerome undertook the translation work
Still, this was a unified church largely within the Roman Empire that was the *direct* hierarchical predecessor of all of the sects mentioned, including the Roman Catholic sect. As the decisions regarding the Scriptures to include were made by officials from several regions, it is not really appropriate for any one of these sects to assume sole credit for those decisions OR for the first bible.
In other words, you have 2 choices
a) it was the early church of the Roman Empire, which predates the Roman Catholic Church
b) it was the hierarchical ancestor of the Roman Catholic Church, and so all of the Eastern Orthodox churches and Oriental Orthodox churches can claim equal credit with the Roman Catholic Church in this matter.
As for "Who organized the Bible in the way it is now?"...
Apparently the very early Greek codices had the same organization for the New Testament as we do today. The Old Testament organization follows the Greek Septuagint, which was translated by Jewish scholars in or near the second century B.C.E.
Luther changed the organization by being the first to include a separate section titled "apocrypha" for those Scriptures that Roman Catholics call "deuterocanonical" and Orthodox call "Anagignoskomena". This was done in the early 1500's, and all English Protestant bibles follow that same practice - either including a separate sections titled "Apocrypha", or omitting those Scriptures altogether.
More about when the additional Scriptures (that is, additional to Hippo/Carthage) were added can be found here.
Interesting: Baruch was not included as late as the 8th century (Codes Amiatinus). I have been unable to determine exactly when it was officially included by name. It *seems* to have b