Sabbath in Islam
The Sabbath, in Hebrew שַׁבָּת (shabat), in Arabic سبت (sabt), is a day when Jews are supposed to take a rest.
And We raised over them the mount for [refusal of] their covenant; and We said to them, "Enter the gate bowing humbly", and We said to them, "Do not transgress on the sabbath", and We took from them a solemn covenant.
The word سبت itself also means "Saturday" in Arabic. 
Muslims have Friday as the day of worship, but Jews observe the Sabbath (=Saturday in Arabic and Hebrew). A hadith claims that the Jews were diverted from Friday into Saturday:
It is narrated by Abu Huraira and Huraira that the Messenger of Allah (ﷺ) said:
It was Friday from which Allah diverted those who were before us. For the Jews (the day set aside for prayer) was Sabt (Saturday), and for the Christians it was Sunday. And Allah turned towards us and guided us to Friday (as the day of prayer) for us. In fact, He (Allah) made Friday, Saturday and Sunday (as days of prayer). In this order would they (Jews and Christians) come after us on the Day of Resurrection. We are the last of (the Ummahs) among the people in this world and the first among the created to be judged on the Day of Resurrection. In one narration it is: ', to be judged among them".
In Quran 2:65 we read that Allah transformed Jews into apes and monkeys because they didn't observe the Sabbath (=Saturday):
And you had already known about those who transgressed among you concerning the sabbath, and We said to them, "Be apes, despised."
So Allah commanded the Jews to observe the wrong day and then punished them when they didn't do it.
In the Bible
The Bible says that they should rest just as god rested on the 7th day after he created the world:
“Remember the Sabbath day by keeping it holy. Six days you shall labor and do all your work, but the seventh day is a sabbath to the Lord your God. On it you shall not do any work, neither you, nor your son or daughter, nor your male or female servant, nor your animals, nor any foreigner residing in your towns. For in six days the Lord made the heavens and the earth, the sea, and all that is in them, but he rested on the seventh day. Therefore the Lord blessed the Sabbath day and made it holy.
Seven days week in Islam
As we've seen, according to the Bible, Jews should take a rest on the 7th day, which is connected to the story of creation where God created the world in 6 days, but rested on the 7th day. So 7 day week makes sense in Judaism. However in Islam, the resting of god is rejected:
And We did certainly create the heavens and earth and what is between them in six days, and there touched Us no weariness.
From Tafsir Ibn Kathir:
Qatadah said, "The Jews, may Allah's curses descend on them, said that Allah created the heavens and earth in six days and then rested on the seventh day, which was the Sabbath. This is why they call it a holiday. Allah the Exalted then sent down denial of their statement and false opinion.
So in Islam, god created the world in 6 days and that's it. There's no reason for week to have 7 days. Yet Islamic week has 7 days. It seems that Muhammad abolished the resting of god, but forgot the connection to the 7 days of the week which is based on the story of 6 days of creation plus 1 day of rest.
There is a fatwa where Islamic sholars were asked why the week in Islam has 7 days, but they didn't have a clear answer:
We cannot be certain about anything with regard of the division of the week into seven days, or the time when this division began – was it before the creation of the heavens and the earth, or was the creation of the heavens and the earth what led to this division? The reason for that is that we have no clear evidence which states anything certain to that effect..
- Tafsir al-Jalalayn explains the verse as "`do not act unjustly in the Sabbath` by fishing during it"