Is there a scriptural basis for treating the non-Gospel books of the New Testament as scripture? Without citing any books of the New Testament besides the Gospels as evidence (since that would be self-referential), is there a scriptural basis (either in the Gospels or in the Old Testament) for treating any book of the New Testament besides the Gospels as scripture rather than as supplemental history/commentary/scholarship in the tradition of the Jewish Talmud?
Is there a scriptural basis for treating the non-Gospel books of the New Testament as scripture?

Short answer: yes.
Jesus commissioned 12 of his disciples as 'apostles' and gave them authority to teach. If you accept Jesus' words as scripture (as recorded in the gospels), you should accept the words of his apostles as scripture as well, as He said to them:

18 ...“All authority in heaven and on earth has been given to me. 19 Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, 20 and teaching them to obey everything I have commanded you. And surely I am with you always, to the very end of the age.” - Matthew 28:18-20 NIV (emphasis added)
Again Jesus said, “Peace be with you! As the Father has sent me, I am sending you.” - John 20:21 NIV

Jesus specifically authorized His apostles to teach, and to do that in the authority that He Himself had received from His Heavenly Father. For this reason, the early church placed particular importance on what were judged to be authentic writings of the apostles (as well as their companions who were believed to have acted as scribes for them - e.g. Mark and Luke). The gospels themselves actually fit in this category - the early church trusted that the (canonical) gospels were accurate because of their author's apostolic credentials. For this reason, there is no legitimate basis for separating the gospels into a different category of 'scripture' that does not equally apply to the rest of the New Testament1.

1. In broad terms at least. There were some arguments about the authenticity of individual books that later came to be accepted (or excluded) from the New Testament canon. However these arguments centred around whether they were authentic in regards to their apostolic credentials. See http://en.wikipedia.org/wiki/Antilegomena for further information on these disputes.