Does the Bible say that Saturday is the seventh day?

The biblical proof you're asking for doesn't exist. One cannot determine by the Bible alone that the day we call "Saturday" is the seventh day of the biblical week.

Make no mistake, the Bible is God's unique revelation to mankind and, therefore must be given first place in our quest for truth, but Scripture nowhere teaches that it is the sole source of all true knowledge.

That being said, we can know that our "seventh day," the day we call "Saturday," is the seventh day of the God-instituted biblical week because it has been continuously observed in some manner not only by the Jews but also by historic Christianity for literally centuries. The early Christians recognized the Sabbath, and so have Christians across the centuries. The Orthodox Church has always referred to our Saturday as "the Sabbath," as they still do today. The Orthodox still follow the ancient tradition of refraining from fasting on Saturday because that day is the "Creation Sabbath" and ought to be honored as such.

In addition, the writings of the early church fathers clearly distinguish between the Sabbath and Sunday, stating in the plainest language that the day we call Sunday is the first day of the week. That understanding was preserved through the centuries-old traditions and customs of historic Christianity. These ancient faith traditions have been preserved by Eastern Orthodox and Roman Catholic communities alike, as well as by the Ethiopian and Coptic Christian communities. It is utterly impossible that all these centuries-old faith traditions could have lost sight of which days were the first and seventh days of the biblical week.

And then there are the Jews that have been scattered all over the world for many centuries. One could easily believe that a single Jew could wake up one Friday morning and mistakenly think it was Saturday, but it's not conceivable that this could happen to millions of them scattered over many countries. Because of the witness of the faith traditions of both Jews and Christians, the whole world today knows that the day we call Saturday is the seventh day of the week and that it's traceable all the way back to the time of Jesus.