There is no way to know for certain.
However, there are a few indications that what we have the "correct" day.
Firstly, there is the halacha that if you do not know what day it is, you keep Shabbat on the 7th day from the last time you kept shabbat. So even if the calendar did a shift, presumably the Jewish people would have shifted the name of the correct day.
Say, for example that next year the world Calendar people decided to move the week 3 days. Then from that time on, "Shabbat" would be on Tuesday, and not on Saturdays. In other words, there is no direct connection between Saturday and Shabbat, except for the fact that they currently coincide.
Now, Historically, the first Shabbat was only kept after the Israelis left Egypt, and we are told by Moses that this day is to be the day of rest. We must assume, but we do not know for a fact, that Moses told us the correct day according to Gd's will.
Lastly, even today in Israel, the days of the week are named "day one, day two, day three, day four, day five, day six, Shabbat" They are not given any other names.
The aspect of your question regarding when Saturday became Saturday, and Sunday, Sunday is a good one for the History stack exchange, but I don't think would be well answered on the Judaism one.
One last point. In America, some calendars show Monday as the first day of the week, and other calendars show Sunday as the first day. It is interesting to me, that on a Christian theological level, Monday should be the first day, and Sunday the last day. Also from a secular, "business" perspective, Monday should be the first day and Sunday the last. However for Jewish people, Sunday being the first day of the week is the most natural. My point here, is to restate what I said earlier, that what name we give the 7th day is not as important as the fact that it is the 7th (i.e. Shabbat) In Israel, Sunday is a workday and the first day of the week.
After looking some things up on Wikipedia, I noticed that the ISO sets Monday as the 1st day of the week, and Sunday as the 7th. So indeed the "world Calendar folks" have said that the 7th day is Sunday, and yet Jews still keep the 7th day (Shabbat) on Saturday.