I am not sure where you got “him fulfilling” from: the printed text of the play has “fulfilling him”.
In this speech Salieri personifies the music (the Adagio from Mozart’s Serenade No. 10) as expressing a “need” which is “unfulfillable”, that is, a desire which cannot be satisfied. The music “fulfills him who hears it, utterly”, that is, it is completely gratifying for Salieri to listen to:
Salieri: [… The oboe] hung there unwavering, piercing me through, till breath could hold it no longer, and a clarinet withdrew it out of me, and sweetened it into a phrase of such delight it had me trembling.
Peter Shaffer (1981). Amadeus, p. 18. New York: Harper & Row.
Why Salieri says that the Adagio expresses an unfulfillable need is not clear at first. The movement has a lyrical oboe solo, to be sure, but resolves itself fairly conventionally with a perfect cadence and an E♭ chord (Mozart’s favourite key, symbolic of the Trinity as it has three flats). But in the next scene we get:
Salieri: I ran home and buried my fear in work. More pupils—till there were thrity and forty. More committees to help musicians. More motets and anthems to God’s glory. And at night I prayed for just one thing. “Let your voice enter me! Let me conduct you! … Let me!”
Shaffer, p. 19.
So it is really Salieri who has the unfulfillable desire, that is, to write music as beautiful as Mozart’s, and in the earlier speech he is projecting this feeling onto Mozart’s Adagio.