There does seem to be some connection, but not enough to promote legalism. There is no mention of this relationship in any Jewish-English Concordance, like Strong's.
[Actually, the number seven is held as the "Number of God" in Jewish mysticism. therefore, both Sabbath and seven MAY refer not to numerical values, but to holiness, with "Shabbath" being a day of holiness, and "sheva" being the number of holiness. Consider Jesus' direction, "“I tell you, [to forgive] not seven times, but seventy-seven times." ]
The word "sabbath" is used to indicate a TIME PERIOD (the seventh day, or even the seventh year) that is to be held Holy to The Lord.
But this term can be applied to ANY DAY at ANY TIME, without reference to "seven" of anything. For example, The Festival of Tabernacles and The Passover are ALSO called "sabbaths", although they do not necessarily occur on Saturday.
In fact, in John 19:31, we learn of a "Special Sabbath", which was apparently a mid-week "holy day". [Some theologians believe that the "Special Sabbath" was The Passover, and was on WEDNESDAY that year -- thus supporting the idea the Christ was crucified on WEDNESDAY, not Friday.] this idea of a mid-week Sabbath indicates that a Sabbath DOES NOT have to be on Saturday!!!
Continuing with the idea that both "Sheva" and "Shabbath" contain the idea of holiness, we can see that it is not really important what DAY you hold as the Sabbath, but it is much more important that you indeed DO HAVE one day as a Sabbath. And THIS would be more in line with Jesus' teaching of "fulfillment of the law" rather than strict, legalistic [and potentially too narrow] interpretation of The Law.