Deuteronomy 10:8-9 states:

At that time the L-RD set apart the tribe of Levi to carry the Ark of the L-RD’s covenant, to stand in attendance upon the L-RD, and to bless in His name, as is still the case. That is why the Levites have received no hereditary share along with their kinsmen: the L-RD is their portion, as the L-RD your God spoke concerning them (JPS translation).

Rambam, in his Mishneh Torah, at the end of the Book of Zera’im, at the conclusion of Hilkhot Shemittah ve-Yovel (13:13), makes the point that the Torah’s aim regarding the tribe of Levi need not be confined to the Levites themselves: 

Not only the tribe of Levi but every single individual from among the world’s inhabitants whose spirit moved him and whose intelligence gave him the understanding to withdraws from the world in order to stand before God to serve and minister to Him, to know God, and he walked upright in the manner in  which God made him, shaking off from his neck the yoke of the manifold contrivances which men seek- behold, this person has been totally consecrated and God will be his portion  and inheritance forever and ever. God will acquire for him sufficient goods in this world just as He did for the priests and Levites. Behold, David, may he rest in peace, says: L-RD, the portion of my inheritance and my cup, You maintain my lot (Psalms 16:5) (translation found in Professor Yitzhak (Isadore) Twersky, A Maimonides Reader [West Orange, New Jersey, 1972], p. 139).

Why did the Rambam see fit to cite the verse in Psalms in this context? I believe the answer is as follows:

In a Hebrew article, “Did Rabbi Abraham ibn Ezra Influence the Rambam?” in the volume Rabbi Abraham ibn Ezra: Studies in the Writings of a Twelfth Century Jewish Polymath, edited by Isadore Twersky and Jay M. Harris (Cambridge, 1993), Hebrew section, pp. 21-48, Professor Twersky demonstrated various cases where a Maimonidean idea is already present, in some form or another, in the works of the ibn Ezra. This insight is a key that can unlock further correlations between the two thinkers.

The ibn Ezra lived from 1092-or 1093 until 1164 or so. Rambam’s dates are 1138-1205. Pursuing the of connections between the two figures further, and searching for other correlations (if not direct influences), if one looks at the commentary of the ibn Ezra to our verse in Parashat ‘Eqev (Deuteronomy 10:9), one finds the following: Commenting upon the phrase the L-RD is their portion, ibn Ezra cites the same verse from Psalms that Rambam does at the end of Hilkhot Shemittah ve-Yovel, citing King David’s affirmation that God is his inheritance! R. Abraham ibn Ezra adds that the (true) service of God consists of understanding His ways.

The verse in Psalms expands the field depicted in the verse in Deuteronomy. R. Abraham ibn Ezra utilized the verse from Psalms (which depicts King David- from the tribe of Judah, not a priest or Levite who had exclusive rights in the bet ha-miqdash) as evidence that anyone (even a non-Jew!) can choose to take God as his “inheritance” and totally consecrate himself to God. This privilege is not limited to priests or Levites. Perhaps Rambam, for his part, used the verse regarding King David, with its suggestive Hebrew word of gorali (my lot, my inheritance) in the same manner as well.

Both Rabbi Abraham ibn Ezra and Rabbi Moses Maimonides saw our Parasha’s depiction of the consecration of the Levites as expressive of a standard of devotion to god that is, ultimately, within the reach of all.