No, none of the NT relies on it’s own statement of scriptural validity. The closest is the line in 2 Tim 3:16. It is used most often by evangelicals to prove the authority of all scripture as we have it now, OT and NT. In fact, Paul nowhere claims that his writings are new additions to scripture, though he does take the time to correct and teach as an apostolic authority, which later, as the Church reads him, finds authority as “scripture”. But for Paul, 2 Tim 3:16 is focused on the OT, which even at that point in his writings was not fully canonically established; only after the fall of Jerusalem did the surviving Jewish community authorize what we know as the OT, though they accepted various texts under the form of Pentateuch, Prophets, and Writings.
The same is true for all other books except for possibly the Revelations of John. All other books went through a process of development and acceptance or rejection by various parts of the Church until the Council of 384, at Hippo, I believe (correction welcome) when the present NT canon was set. And the Christian, version of the OT, with the Deutero-Canonicals, was accepted. But nowhere does any book, of its own accord call itself scripture, that is left to the larger ruling community to decide, and that took centuries.