No, not usually. Before the modern period, they were stallions, and so willing to tear into one another. Blood would not be an automatic freak-out to them. Warhorses were chosen for aggressive personalities and courage.

Hunter's mounts and pack horses today have to be inured to the smell of bloody carcasses, which is done by getting them used to having raw meat around. The mounts have to learn to stand steady and not take off when someone is shooting from the saddle. All this is usually done by gradual introduction.

The same applied to young warhorses. They were used to trumpets and yelling, they learned to tolerate the motion of the Saracen close to them, and finally to take the shocks and knockdowns of jousting. They were used to being in the middle of friendly melee, with shields moving beside them and wooden swords and axes flailing around them.

Ann Hyland, in Training the Roman Cavalry, from Arrian's Ars Tactica (Sutton Publishing, Grange Books, London, 1993) broke down just what he was doing to reach the conclusion that his exercises were particularly pointed at restoring the confidence of horses that had been injured in combat. They learned to trust their riders again, riding headlong at other horses in close passes, until they could remember that, no, this doesn't always get you hurt.

Follow-up question: How were horses desensitized in general to the loudness and violence of war?