Is the European war tradition really more focused on death than others?

No. Rarely was the goal of Western warfare to kill the enemy - the goal was usually to capture a fortress or city, or to compel the enemy to negotiate on one's preferred terms. Killing the enemy was simply one of the paths to those goals, not an end in itself. For much of European history, sieges were a key element of warfare, and the focus was on taking possession of the fortress or city. If the defenders were willing to surrender, they would often be allowed to leave with the weapons, and sometimes as much else as they could carry.

Western commanders often avoided open battle (the side expecting to lose instead retreating, often into the "safety" of a fortress). The condottieri, mercenary commanders in Italy, were famous for a very cautious and low-casualty approach to battle. Knights were regularly taken prisoner for ransom rather than killed. Defeated mercenaries were often recruited rather than killed.

There are many examples of battles where European armies were crushed, with overwhelming casualties, when fighting non-European foes. Non-European states generally aimed for the same goals as Europeans: taking or protecting fortresses or cities, and compelling the enemy government to negotiate on desirable terms. As with European states, killing the enemy was a way to try to achieve this, and non-European states did so, sometimes with spectacular success. For example, many of the crushing defeats of Roman armies were inflicted by non-European powers - Hannibal, born in North Africa and serving a state that began as a colony of an Asia state, annihilated a Roman army at Lake Trasimene, and killed about half of the enormous Roman army at Cannae; Crassus and his army met their end at the hands of the Parthians, etc. This didn't end with the end of antiquity. The Mongol invasions of Europe didn't display European aiming-to-kill against Mongol aiming-to-intimidate, nor did wars against the Ottomans, Zulus, and many, many more. Even into modern times, we have examples of bloody warfare between the West and non-West, e.g., in the Pacific War in WWII, the Korean War, and even into this century in the fighting in Afghanistan and Iraq.

So far, I have largely consider warfare involving organised state vs organised state. What about "primitive" warfare? Sometimes - often even - the goal was to kill the enemy. Sometimes, the aim was to take heads for status, magic power, the demonstration of courage. The goal might be thoroughly genocidal: kill all enemy adult males. Sometimes the goal was simply to eat the enemy. Particularly brutal examples of warfare could combine these.

There is plenty of commentary online on "the Western Way of War", much of it driven by Victor Davis Hanson's book, The Western Way of War: Infantry Battle in Classical Greece (Alfred A. Knopf, 1989). Hanson claimed that ancient Greek warfare was characterised by seeking decisive battle (and avoiding deception), while non-Western war was characterised by sneaky deception and avoidance of battle. Further, Hanson claimed that this pattern continued from the ancient period through to today. This is a different dichotomy from that given by your professor, but it is related in that decisive battle can lead to much killing. I would describe Hanson's book as very poor history - his argument depends on cherry-picked examples, ignoring European examples of deception and avoidance of battle and non-Westen examples of seeking decisive battle. This extremely poor treatment of supporting evidence by Hanson is of course widely discussed in negative commentary on his work. Some conveniently-online examples of the discussion:

• https://sites.psu.edu/hoplitewarfare/a-western-way-of-war/

• https://sites.psu.edu/sarahreeseeportfolioeng202a/greek-hoplites-and-thewestern-way-of-war-2/

• https://www.ancientworldmagazine.com/articles/western-way-war/

As already said, this isn't your professor's dichotomy, but the discuss should make it more clear that there was little difference between Western and non-Western warfare in principle (in practice, there were differences during the time that the West had breech-loading rifles, modern artillery, machine guns, etc. and the non-West was largely without these).

I would be inclined to say there is no "Western European way of war". Or, more precisely the way of war was as fluid and changed just as easily in Western Europe as elsewhere in the world.

There were European cultures where taking prisoners was not in fashion in certain periods, but I would say that is rather the exception and not the rule.

For example in Roman and Hungarian history (the two subjects I am most familiar with, though I realise Hungary does not qualify as Western-Europe) prisoners were seen as a source of income. In the campaigns of Rome, prisoners, who could be sold as slaves were often a substantial part of the booty. In Hungary, during the Ottoman wars, taking a high ranking Turkish soldier as prisoner was not just a source of income, but also good for one's career advancement.

That is not to say there are absolutely no examples of "taking no prisoners" policies in European history.

E. g. the soldiers of the Old Swiss Confederation were famous for having a "no prisoner" policy in the XIVth century, but AFAIK the reason for this was precisely that taking prisoners of war was a widespread custom in contemporary Europe and a very lucrative business. From what I read the citizens of the first cantons were afraid that the great sums that were involved would break down discipline in the ranks and would be cause for strife after the battles. So they did not take prisoners at all, to be on the safe side.

Now, for the conquistadors, I rather doubt that Spanish military culture was against taking prisoners. I have to admit that I am not very familiar with XV-XVIth century Iberian military culture, but I think the maxim that taking wealthy prisoners is a way to get good money would hold true for them too. So why did they not take prisoners in their wars against native Americans?

Well, first of all, I am pretty sure they actually did. Both Cortez's and Pizzaros's strategy was based on capturing the head of state and securing as many aristocratic hostages as they could... So the statement that they did not take prisoners seems really strange.

I would say that the Spanish reputation for "taking no prisoners" probably comes from their war against the Tlaxcalans, the battle of Otumba and the Siege of Tenochtichlan.

In the first case the Spanish forces were horrendously outnumbered, in a very dangerous situation, basically under siege. Obviously, they did not take prisoners, because they needed every single soldier in the line and could not waste one man on guard duty.

In the battle of Otumba, the Spanish were in an even worse situation, if you can believe. They were a beaten force, cut off from supplies and reinforcements, desperately trying to break out of encirclement. Again, not an ideal situation to take prisoners.

In the last case I mentioned, in the Siege of Tenochtitlan, the situation was completely different. The Spanish had many Indian allies now, they had secure supply lines and a clear way to retreat if things went wrong for them. So why were they not taking prisoners?

Again... in the end the Spanish did take the Mexica leader Cuauhtémoc prisoner, so obviously some caveat must be applied.

But indeed, for the common Mexica soldier (or civilian for that matter), there was no quarter given during the siege. Well, if we are to believe Cortes's own letters, he actually planned to take the surrendering enemies as prisoners, but as it so often happens during sieges, discipline broke down and the soldiers went on a spree of loot and murder.

Cortes's grasp must have been tenuous even over his Spanish troops in the desperate house-to-house fighting that characterised the late phase of the siege. And over the Indian allies, who gave the overwhelming majority of the besieging forces, he was only ever nominally in command. According to the letter Cortes wrote Charles V, the cruelty of his allies horrified even him, and he was not a squeamish person.

On a side note, it might be an interesting question whether anyone who lived through the siege of Tenochtitlan in 1521 was there at the sack of Rome six years later in 1527. Both atrocities were committed by troops nominally fighting for the cause of Charles V, and in both cases the discipline broke down because of the lack of proper leadership - in Rome the commander died during the assault and in the case of Tenochtitlan there wasn't any proper leadership or discipline to begin with. It would be interesting to see how these two events compared to an eyewitness who lived through both.

I think I will close my ramblings with this, and try to summon someone who knows more about the conquest of Mexico, like u/400-rabbits and he will hopefully correct me if I told something really stupid.

EDIT: You might want to xpost this to r/warcollege people there are very well-versed in the history of Western European warfare, so they might give you a more holistic answer regarding that topic. Also, it is a much smaller sub than this one, so you will get an answer with bigger probability.

Besides the answers already here, I also want to point out that European warfare was as much about intimidation as everyone else.

For the most part, the goal of battle in general to intimidate the enemy into running away or surrendering. Europe was in no way the exception. As /u/Iphikrates outlines here and here, the way the famous Spartans took great steps into intimidating their enemies, including through propaganda and dress, to make their enemies run away or even not fight at all. Against enemies who don't get intimidated, the Spartans don't actually have any exceptional track records.

This is not any different in the other time periods. For instance, the cavalry's role was very much or perhaps even more about scaring the infantry into running away. And even during the heyday of the knight, infantry that could stand their ground could prevail against the knight, courtesy of /u/Hergrim. And intimidation was a large factor why the Spanish horsemen were (initially) effective against the Mesoamericans who had never seen horses. The famous polish winged hussar's wore their wings to appear larger to scare the enemy. Various early-modern grenadiers wore tall bear-skin hats to scare their enemy. Even the basic tactics of warfare like ambushing and attacking the flanks and rear were very much to intimidate the enemy into running, by presenting the battlefield equivalent of a jump-scare. And formations like the deep column or the square were adopted to make the men feel safer and prevent them from running away (if only by physically blocking them from running), not to kill more of the enemy.

Finally, I need to mention that while the conquistadors punched above their numerical weight during the conquest of what is now Latin America, it was the native polities themselves who did the heavy lifting in that conquest, not the conquistadors.

Your professor is exceptionally r***rded if he thinks Spanish conquest was the violent one. Yes, go find how many native NorthAmericans are left in the USA (barely any). Now go find how many native LatinAmericans are left (all of them). Mexican indigenous tribes were very violent and they actually killed their fellow native countrymen with special cruelty. Go tell your professor that, and also to learn history.