Firstly, because Muslims are not Jews subject to the mitzvah of the Mosaic covenant, of which sabbath observance is one of the most important religious laws.
Friday is, instead, prescribed in the Qur'an as the day of prayer (not rest, there is no day of rest in Islamic theology so far as I am aware), when Muslims are to cease from trading when they hear the call to prayer and then resume their labours to spread the bounties of God, after the
Jumu'ah at the Mosque:
O you who have faith! When the call is made for prayer on Friday, hurry toward the remembrance of God, and leave all business. That is better for you, should you know. And when the prayer is finished, disperse through the land and seek God's grace, and remember God greatly so that you may be successful.
— Qur'an, Surah Al-Jumu'ah (62), Ayahs 9-10
The fact that the Qur'an assumes, from the outset, that people will be "
at business" on Friday when the summons to the prayer sounds, is reflective of the fact that Islam treats Friday - yes as the holiest day of the week but - not as a "
sabbath" day of rest akin to the Jewish Shabbat on Saturday and the Christian Sunday.
And that's because the underlying theological basis of the Sabbath in the Hebrew Bible (that God rested from his labours after the six days of creation) appears not to be present or emphasized in the Qur'an.
Secondly, there is a Hadith which further substantiates the Qur'anic designation of Friday as the Day of Prayer, by arguing that Friday was actually the original, primordial holy day that Islam has restored to its obligatory status.
Nevertheless, God had according to this tradition still directed Jews and Christians in prior ages to worship on Saturday and Sunday, respectively:
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".
Sahih Muslim: Book 4, Hadith 1862