Why (how) does Jesus' coat have no seam? When Jesus is arrested and His captors take His clothes, they split them in four parts. However, His coat they did not split, as John records, "the coat was without seam, woven from the top throughout." See John 19:23,24.

Is this seamless robe only significant because it led to the fulfillment of the scripture in Psalm 22? Is there a theological or doctrinal symbolic significance of having no seam? 

I don't even know the process of weaving - how would you make a coat without seams? (I assume it has sleeves)
There is definitely theological significance to the seamless tunic of Jesus, "woven from the top throughout."  Here is a linguistic argument for that significance.

The Greek for John 19:23 (Nestlé-Aland) says εκ την ανωθεν υψαντος δι ολου.  ek ten anothen hupsatos di holou   "From the top woven by entirety", literally.

That same word ανωθεν (anothen, from the top) happens to show up in John 3:7, which says in the KJV, "Marvel not that I said unto thee, "Ye must be born again." The Greek phrase is δει υμας γεννηθηναι ανωθεν.  It-is-necessary you will-be-born again? top?.

Jesus's garment is woven from the top in just one piece. Yet we must be born (again?) (from above?)  Is it possible that the seamless wholeness of the garment is a symbol of the wholeness conferred by baptism?