When we look for the messianic prophecies that Jesus completed, should we only look at the Old Testament? What would be the basis for using the prophecies which were in books written after his death?
Usually when people use the term "Messianic Prophecy," they're referring to a prophecy concerning the "Annointed One" (messiah) that was to come. (Here's a list of 45+.) These prophetic predictions/descriptions were about some yet unknown person to come.
When Jesus speaks of himself, it could certainly be described as prophecy, it doesn't exactly fit the pattern of previous "messianic prophecies."
The OT Messianic prophecies are another assertion of the validity of Jesus, because these predictions were made and recorded hundreds of years before Jesus was born.
Jesus' predictions about himself don't provide the same kind of external proof because to assume they were accurate presumes that the author is recording the truth, and that presumption is essentially assuming the conclusion that Jesus is the Messiah. A pessimistic analysis of these predictions might say that they were fabricated after Jesus came and so, of course they are accurate. Whereas the Hebrews had spent the previous 400+ years studying the predictions made before Jesus' fulfillment of any of them. The accuracy of those predictions serves as proof that Jesus' coming bears the stamp of God.