Matt 4:4 by itself does not prove Sola Scriptura. Keep in mind, though, that Jesus (and others) would quote Scripture (“it is written”, “Have you not read what God has said”, etc.) as the final proof of what they were teaching. I do believe Jesus gave us a way to test “traditions” with His example of the Corban rule (see Mark 7). I know the argument - “Sure, those are human traditions, but our Traditions are from God”. But have you ever tested your traditions by this method?
Just a thought, it is written, ‘What then? Are we better than they? Not at all; for we have already charged that both Jews and Greeks are all under sin; as it is written, “There is none righteous, not even one; There is none who understands, There is none who seeks for God; All have turned aside, together they have become useless; There is none who does good, There is not even one.”’ (Rom 3:9-12 NASB). There is an exception to the “all have sinned”, and that is found in 2 Cor 5:21 (He made Him who knew no sin to be sin on our behalf, so that we might become the righteousness of God in Him.), which is speaking of Jesus. The Scriptures make no other exception for being without sin. Yet, the Catholic Church says, "The most Blessed Virgin Mary was, from the moment of her conception, by a singular grace and privilege from almighty God and by virtue of the merits of Jesus Christ, Saviour of the human race, preserved immune from all stain of original sin (CCC 491). How is this tradition made dogma NOT contrary to the written Word of God? (Just something to ponder, as this is not the topic of the thread.)
In conclusion, Matt 4:4 by itself does not prove Sola Scriptura, but when you examine all of Scripture, it’s nearly impossible to deny Sola Scriptura (this was a major factor in my leaving the Roman Catholic Church). For the record, I was born and raised Roman Catholic, and had received 6 of the 7 sacraments by the time I was 30. God has delivered me out of that religious system, and I am now a reformed Baptist.