The quick answer is that cavalry during the American Civil War (ACW) rarely attacked massed, set infantry formations that were dug in and/or expecting an attack. This is the only scenario where using caltrops would have been effective because, presumably, the people laying these things out would be doing so because they expected a cavalry attack to come from one particular direction.

But that's not really how cavalry worked in the ACW. Early in the war, the Union doctrine held that cavalry should be used for scouting, and to screen infantry movements (so ride alongside and ahead of infantry formations to fend off enemy cavalry scouts, or warn of ambushes). It was the Confederacy that took cavalry tactics to a whole new level early in the war, using large bodies of cavalry to plunge into enemy territory disrupt communications, destroy supplies, and pick off smaller/unprepared bodies of infantry (the Union eventually caught on and adopted this as well).

Now, you might ask why these smaller bodies of troops didn't lay down caltrops to defend against just such an attack. But to be protected, you'd have to lay a ring of these things 360 degrees around your whole camp or formation, then pick them up each morning (and hope that your own pack mules and cavalry don't step on them). An attack could literally come from any direction so it would be impossible to know where to lay these out except in all directions.

In battle, there would be no need to lay these caltrops in front of your lines, as cavalry very rarely charged mass infantry head-on (it was suicide to do so unless the cavalry vastly outnumbered the infantry formation). You might wonder why you wouldn't lay caltrops along your rear or flanks, yet lines of battle were often so fluid that to lay them down in one area would mean little, as you would likely not be in that same place even an hour later to enjoy their protection.

And even if you were in a set position, and didn't move, and you wanted to protect your flanks from attack, if a cavalry formation could get behind you, or at an exposed flank, so could an infantry regiment, and they would not be too bothered by the caltrops. It was always wiser to secure your flanks with infantry support and cavalry screening. So, your best bet if you really wanted to protect against a cavalry attack was to tie in with other infantry bodies, and to have your own cavalry nearby to screen and protect you. If you do get surprised: mass, turn, and fire before they are on top of you. If you are too disorganized, spread out, or with too little time to react in such a manner, you definitely would not have had time to lay down caltrops.

I suspect this is why we don't see anything about them in the ACW historiography. Although obstacles to hinder and injure charging infantry did indeed exist (and in abundance), such preparations just weren't practical for cavalry at that time.