The one question I have seen hinted at but not directly asked (unless I missed it) is: were all of the glued faces properly face jointed flat and square *before* the glue up? Then, were each of the now 2x4 sections face jointed before they, too, were glued together to form the final 4x4?
I ask the question because if someone is using 2x2's where most of us would have likely used two 8/4 boards that were cut into 2"x4" pieces for the glue up, then this may very well be store-bought S4S stock (such as that sold by the BORGs) that may or may not be perfectly flat and square by the time it is ready for use (and more often than not isn't). It is also difficult to precisely align four 2x2 sections during glue up, so assembling them into 2x4 pieces, face jointing the 2x4 sections to remove any alignment errors, then gluing the two 2x4 sections together -- and, if needed, face jointing and thickness planing the fully assembled 4x4 to eliminate any error and hide your glue lines.
If the face jointing was skipped, then it is a bit more iffy as to the quality of the glue lines because there is a far greater likelihood of voids within the glued assembly. If so, unless a good gap-filling glue (such as epoxy) was used, there will be weaknesses within the boards that may, or may not, allow for future delamination depending upon the forces being exerted on them. (For instance, a horizontal bolt with a heavy vertical load that is inserted dead-center in the vertical 4x4's will be concentrating it's vertical load in a manner that could cleave a weak glue line -- something you would not need to be concerned with if it were a strong glue line.)
Otherwise, if all faces were properly jointed and clamped tightly during the glue up, there there is little reason for concern so long as you were using a good-quality modern glue (which Titebond-III is).
If you question the quality of the glue up, then you might want to screw them together (you would need four screws each joining one 2x2 to its neighbor at 3, or so, points along the post's length). Ideally you would recess your screws so that you could then use grain-aligned tapered plugs to permanently hide them from view. If the plugs are cut from extra boards from your stock and chosen for matching color and grain similarities then the plugs will all but disappear from view when you are done.