No, it proves nothing (!) and mostly (but not exclusively though) because it has no provenance. For any given antiquity, provenance is essential for credibility and historicity. That of this ancient manuscript is as yet undetermined.

[Emphasis mine.]
A discovery by a Harvard researcher may shed light on a controversial aspect of the life of Jesus Christ.
Harvard Divinity School professor Karen L. King says she has found an ancient papyrus fragment from the second century that, when translated, appears to indicate that Jesus was married.
The text from the New Testament is being dubbed “The Gospel of Jesus’ Wife.” The part of it that’s drawing attention says, “Jesus said to them, ‘my wife’” in the Coptic language. The text, which is printed on papyrus the size of a business card, has not been chemically tested to verify its dating, but King and other scholars have said they are confident it is a genuine artifact.
“Christian tradition has long held that Jesus was not married, even though no reliable historical evidence exists to support that claim,” King said at a conference in Rome on Tuesday. “This new gospel doesn’t prove that Jesus was married, but it tells us that the whole question only came up as part of vociferous debates about sexuality and marriage. From the very beginning, Christians disagreed about whether it was better not to marry, but it was over a century after Jesus’s death before they began appealing to Jesus’ marital status to support their positions.”
King, who focuses on Coptic literature, Gnosticism and women in the Bible, has published on the Gospel of Judas and the Gospel of Mary of Magdala. She presented her research Tuesday evening in Rome, where scholars are gathered for the International Congress of Coptic Studies…
Further,
The fragment has eight incomplete lines of writing on one side and is badly damaged on the other side, with only three faded words and a few letters of ink that are visible, even with the use of infrared photography and computer-aided enhancement.
The private owner of the papyrus first approached King in 2010. King said she didn’t believe the document was authentic, but the owner persisted. She then asked the owner to bring the papyrus to Harvard, where she became convinced it was a genuine early Christian text fragment. Along with Princeton University professor Anne Marie Luijendijk and Roger Bagnall, director of the Institute for the Study of the Ancient World, King claims to have confirmed the document is real. The document’s owner has not been named and King said he does not want to be identified.
It’s unclear when the text was initially discovered…
There is more here.
Watch as ignorant, dilettantish, sensationalist media mongers run riot with the above story and the unsubstantiated claims made in it over the next couple of days… And when you do, please remember it is an undated, incomplete, unprovenanced, fragmentary document that can be saying and meaning anything (most likely, that someone centuries after the time of Jesus was of the opinion that he may have been married). Archaeologically speaking, there is simply far too much being read into this not-so-new discovery. Moreover, it is a well-known fact that the Coptic and Gnostic writings of that period are particularly notorious for being spurious and palpably fake.
Now all that said, Dr King is a fine scholar with a good reputation. Her forthcoming journal article for the Harvard Theological Review can be read here.
See also Bible Places:
The problem with today’s headline story is not the discovery of an ancient document that suggests that someone once believed that Jesus had a wife. There were many false and unbiblical teachings in ancient times, just as there are today. The problem is the media can very easily make a minor story into something sensational that appears to threaten historic Christianity…