Were the actions of the 3rd Duke of Alba in the 16th Century considered a Catholic Crusade?

No, it would not be considered in any historical sense a crusade. There hadn't been a crusade (i.e. 1-7) for 300 years at that point. Catholic historians consider the fight against the Albigensians and the struggle to reclaim Andalusia "crusades" ¹. But this struggle doesn't have the hallmarks of a crusade. Namely, a concerted force of Christian soldiers, knights and nobles from all of Christendom fighting against a common foe at the behest of the Pope.

The curious thing about this point of "history" is that it doesn't seem to be remembered in the same way that the massacre of the 2000 Hugenots in Paris a few years later.²

And, if this were a real inquisition, not just a Protestant revisionist historical tract - as James Wylie is famous for, it would have noted that the Inquisition didn't operate this way until an individual backslided on their repentance.³

Furthermore, unlike the Plantagenet kings of England and some of the Emperors of the Holy Roman Empire, the Spanish kings were subservient to the Pope and wouldn't have any authority the kill on person, let alone an entire country on religious grounds. If he ordered the Duke of Alba to do this thing which obviously never happened because it would have at least as much historical implications as the Holocaust and not be overshadowed (as I said above) by the very real massacre of the Hugenots. These things are recorded here, 19th century fake news, futher proof that there is nothing new under the sun.

Even furthermore, 3 years later, Phillip's own half-brother Don Juan of Austria was busy defending Christendom (probably saving the country that Phillip had apparently sentenced to death) at the Battle of Lepanto and that isn't even considered a crusade. ⁴

So, no I don't think anybody would call these actions a crusade.

^{1,2,3,4 All random facts sourced from Light to the Nations part one, The History of Christian Civilization,}

Professor Petrus Johannes Blok was Professor of Dutch History at Leiden University, the Netherlands oldest university, in the late 19th and early 20th century. In 1907 he identified that the decree sentencing almost all Netherlands inhabitants to death, and depriving them of their property, said to have been promulgated in Spain in 1568, was actually a forgery dating from some time in the 1570s. In a sense then, it was, according to the Professor, fake news, but from the 16th century; and generally accepted by historians until the 20th.

See page 42-43 of "Britain and the Netherlands Volume 5" by Bromley and Kossmann .

The Netherlands Inquisition had been running for several decades already and, as Phillip II himself observed, was less merciful than the Spanish. At one time heretics who refused to repent could be burned. If a man or boy repented he could be beheaded, but if a woman or girl repented she could be buried alive.

This article, on the website of St Andrew's University, the oldest in Scotland, quotes estimates of 50,000 to 100,00 deaths. After the Duke of Albi arrived many more thousands of deaths were inflicted, it says.

This was the same Duke of Albi who in 1556 had besieged Rome and threatened to sack the city if the then pope did not accept his terms.

However no attempt was made to exterminate almost all the inhabitants of the Netherlands indiscriminately. The decree, even if it had existed, was not acted upon. That the Duke of Alba did lead a force to the Netherlands and that much persecution followed is true.

The historical details are somewhat disputed and perhaps the History site might provide some answers as to roughly how many perished, and how many suffered other punishment or went into exile.

If, in fact, the Pope had approved a campaign to kill all 3,00,000, many of whom were Catholic anyway, and had called on all Catholics to support it, then that would have been a most extraordinary thing. We can only speculate whether it might have been called a crusade, or what it might have been called.

Rather depressingly, perhaps, what actually happened was, in the context of sixteenth century Europe, more a case of "business as usual".