The seventh day of Creation, God rested, sanctified the day and blessed it, because in it he rested.
God obviously did not need to rest; the logical conclusion is that He did so as an example to man/adam.
It seems as though God continues to operated conscious of a seven day cycle as he calls Noah and his family into the ark, and tells them that in seven days the rain will begin. Sure enough seven days later, the rains begin.
It seems as though Noah, after the flood is conscious of a seven day cycle, for he sends out a dove, exactly seven days later sends out another, and exactly seven days later sends out another. And on that day, God calls him out of the ark and makes an offering upon an altar to God. God smells his offering and is pleased and makes a covenant with Noah and his descendants because of it.
It seems to me that Noah, the righteous man, was keeping Sabbath, and worshiping God on this day. It even seems possible that God was teaching Noah of it before the flood, then after the flood, Noah operates in keeping with it and the worship offered pleases God.
I don't know if there is enough information provided in the text to determine if the rest of the days are evenly divided by seven as well. I provided the text for that as well in case someone understands the calendar(s) alluded to and can figure it.
Note: There are many sevens in the account and it does make one take notice of them and consider it more deeply. This question, though, is based on and pertains only to days.
Do any of the sages expound on this?
Noah and his family enter into the ark--Bereishit 7:1,4
וַיֹּאמֶר יְהוָה לְנֹחַ, בֹּא-אַתָּה וְכָל-בֵּיתְךָ אֶל-הַתֵּבָה: כִּי-אֹתְךָ רָאִיתִי צַדִּיק לְפָנַי, בַּדּוֹר הַזֶּה.
God says in seven days the rain will start--Bereishit 7:1,4
כִּי לְיָמִים עוֹד שִׁבְעָה, אָנֹכִי מַמְטִיר עַל-הָאָרֶץ, אַרְבָּעִים יוֹם, וְאַרְבָּעִים לָיְלָה; וּמָחִיתִי, אֶת-כָּל-הַיְקוּם אֲשֶׁר עָשִׂיתִי, מֵעַל, פְּנֵי הָאֲדָמָה.
Seven days later the rain begins--Bereishit 7:10-11
וַיְהִי, לְשִׁבְעַת הַיָּמִים; וּמֵי הַמַּבּוּל, הָיוּ עַל-הָאָרֶץ.
יא בִּשְׁנַת שֵׁשׁ-מֵאוֹת שָׁנָה, לְחַיֵּי-נֹחַ, בַּחֹדֶשׁ הַשֵּׁנִי, בְּשִׁבְעָה-עָשָׂר יוֹם לַחֹדֶשׁ--בַּיּוֹם הַזֶּה, נִבְקְעוּ כָּל-מַעְיְנֹת תְּהוֹם רַבָּה, וַאֲרֻבֹּת הַשָּׁמַיִם, נִפְתָּחוּ.
On the seventh month seventeenth day of the month the ark rests--Bereishit 8:4
וַתָּנַח הַתֵּבָה בַּחֹדֶשׁ הַשְּׁבִיעִי, בְּשִׁבְעָה-עָשָׂר יוֹם לַחֹדֶשׁ, עַל, הָרֵי אֲרָרָט.
Seven days later--Bereishit 8:10
וַיָּחֶל עוֹד, שִׁבְעַת יָמִים אֲחֵרִים; וַיֹּסֶף שַׁלַּח אֶת-הַיּוֹנָה, מִן-הַתֵּבָה.
Another Seven days later--Bereishit 8:12
וַיִּיָּחֶל עוֹד, שִׁבְעַת יָמִים אֲחֵרִים; וַיְשַׁלַּח, אֶת-הַיּוֹנָה, וְלֹא-יָסְפָה שׁוּב-אֵלָיו, עוֹד.
Offering offered on the altar--Bereishit 8:20:
וַיִּבֶן נֹחַ מִזְבֵּחַ, לַיהוָה; וַיִּקַּח מִכֹּל הַבְּהֵמָה הַטְּהֹרָה, וּמִכֹּל הָעוֹף הַטָּהוֹר, וַיַּעַל עֹלֹת, בַּמִּזְבֵּחַ.