This is the source from the Torah for the mitzvah of Birkat Hamazon. Does this make it a biblical mitzvah?
Please review the following commentaries on the pasuk (Sforno, Ramban, and Chizkuni). According to each one, what is the purpose of Birkat Hamazon?
And it stated, "And you will eat, and be satiated and bless," since you will remember the work of Egypt and the affliction of the wilderness. And when you will eat and be satiated in the good land, you will bless God about it. And our rabbis (Berakhot 48b) received the tradition that this is a positive commandment - and its explanation is that you shall bless the Lord, your God. And so [too], "and you will make a parapet for your roof" (Deuteronomy 22:8); and so [too] "and you will make a Pesach sacrifice to the Lord, your God" Deuteronomy 16:1); [and so too,] "And you will take from the beginning of all the fruit of the land" (Deuteronomy 26:2); and there are many [other examples] like this. And the explanation of "about the good land" is as [if it were written], "and about the good land." He commanded that you should bless Him at every time about the satiation and the land that He gave you - that he gave it as an eternal inheritance to you, and you will be satiated from its goodness. But behold, the obligation of this commandment is in every place.
Note how the Chizkuni breaks down the pasuk into different parts, stating that each phrase in the pasuk corresponds to a different paragraph in Birkat Hamazon.
ואכלת ושבעת וגו׳, “you will eat and be satisfied;” this verse is the one from which the sages derived the first three blessings in the prayer known as ברכת המזון, saying grace after meals. The three pertinent words are: ואכלת ושבעת וברכת “when you have eaten and been satisfied you are to say grace.” The first blessing concludes with the words: הזן את הכל. “He Who gives food to all.” The word את in the line: את ה׳ אלוקיך; this is a reference to the owner of the house in which you have consumed this food, i.e. your host.
הטובה אשר נתן לך, “the good one that He gave to you.” This is a reference to the third paragraph in which we ask G-d to rebuild Jerusalem, compare Talmud tractate Sotah folio 5.
After a person has sated himself, and is revolted by seeing more food, he is obligated to pronounce these benedictions. If this is so, it follows automatically, without having to be spelled out, that when he is hungry and sits down to a meal that he will bless the Lord Who has provided it directly or indirectly. [i.e. a bracha rishona.] He does so by reciting the line concluding with the words: המוציא לחם מן הארץ, “Who has caused the earth to produce “bread,” i.e. food.
According to the Gemara in Brachot, who wrote the different sections of Birkat Hamazon?
Does this make it a rabbinic mitzvah?
With regard to the origins of the four blessings of Grace after Meals, Rav Naḥman said:
Moses instituted for Israel the first blessing of: "Who feeds all," when the manna descended for them and they needed to thank God.
Joshua instituted the blessing of the land when they entered Eretz Yisrael.
David and Solomon instituted the third blessing: Who builds Jerusalem, in the following manner:
David instituted “…on Israel Your people and on Jerusalem Your city…” as he conquered the city,
and Solomon instituted “…on the great and Holy Temple…” as he was the one who built the Temple.
They (the rabbis) instituted the blessing: Who is good and does good, at Yavne in reference to the slain Jews of the city of Beitar at the culmination of the bar Kokheva rebellion. They were ultimately brought to burial after a period during which Hadrian refused to permit their burial. As Rav Mattana said: On the same day that the slain of Beitar were brought to burial, they instituted the blessing: Who is good and does good, at Yavne. Who is good, thanking God that the corpses did not decompose while awaiting burial, and does good, thanking God that they were ultimately brought to burial.
Some additional historical information about the event that prompted the writing of the last bracha:
Read the halacha from Rambam's Mishne Torah. How does Rambam characterize the mitzvah of Birkat Hamazon?
מצות עשה מן התורה לברך אחר אכילת מזון שנאמר ואכלת ושבעת וברכת את ה'' אלקיך ואינו חייב מן התורה אלא אם כן שבע שנאמר ואכלת ושבעת וברכת ומדברי סופרים אכל אפילו כזית מברך אחריו.
Mishneh Torah Hilkhot Brachot 1:1
It is a positive mitzvah from the Torah to bless [God] after eating satisfying food, as it is said: "When you have eaten and are satiated, you shall bless God, your Lord." The Torah itself requires a person to recite grace only when he eats to the point of satiation, as implied by the above verse, "When you have eaten and are satiated, you shall bless...." The Sages, however, ordained that one should recite grace after eating [an amount of bread equal to] the size of an olive.